Efficient Resource As

Crescco CRESCCO IST-2001-33135 Critical Resource Sharing for Cooperation in Complex Systems

Workpackage 1: Eﬃcient Resource Assignment and Communication Protocols in Wireless Networks

Deliverable D1.2 Algorithmic Solutions and Technical Recommendations for Wireless Networks I

Responsible Partner: Computer Technology Institute (GR)

Report Version: 1.0 Report Preparation Date: 15/01/03 Classiﬁcation: PUB Contract Start Date: 01/01/02 Duration: 36 months Project Co-ordinator: University of Patras

Partners: Computer Technology Institute (GR) University of Geneva (CH) Centre National de la Recherche Scientiﬁc (F) Universite de Nice-Sophia Antipolis (F) Christian-Albrechts-Universitaet zu Kiel (D) Universita degli studi di Salerno (IT) Universita degli studi di Roma “Tor Vergata” (IT)

Project funded by the European Com- munity under the “Information Society Technologies” Programme (1998-2002) Contents

1 Frequency Assignment in Radio Networks 3 1.1 The Radiocoloring Problem ...... 3 1.2 Radiocoloring general and planar graphs: Complexity and approximations 5 1.3 Eﬃcient coloring of Squares of Planar Graphs ...... 10 1.4 Radiocoloring Periodic Planar Graphs: PSPACE-completeness and appoximations ...... 13 1.5 Radiocoloring Hierarchically Speciﬁed Planar Graphs: PSPACE- completeness and appoximations ...... 17

2 Fault Tolerant Broadcasting, Energy Consumption and Connectivity of WNs 25 2.1 Fault Tolerant Broadcasting in WN ...... 25 2.2 Energy consumption and connectivity of WNs ...... 26 2.3 Optimal F-Reliable Protocols for the Do-All problem on Single-Hop Wire- less Networks ...... 28 2.4 Distributed Broadcast in Radio Networks for Unknown Topology . . . . . 28 2.5 The Minimum Range Assignment Problem on Linear Radio Networks . . . 29 2.6 The Minimum Broadcast Range Assignment Problem on Linear Multi-Hop Wireless Networks ...... 29 2.7 Energy Eﬃcient Broadcasting in General Wireless Networks ...... 29 2.8 Some Recent Theoretical Advances and Open Questions on Energy Con- sumption in Ad-Hoc Wireless Networks ...... 30 2.9 On the Approximation Ratio of the MST-based Heuristic for the Energy- Eﬃcient Broadcast Problem in Static Ad-Hoc Radio Networks ...... 30 2.10 Minimum-Energy Parameterized Problems in (Wireless Network) Graphs . 31

1 2 Chapter 1

Frequency Assignment in Radio Networks

In this chapter we present the results achieved in the project regarding the Frequency Assignment Problem (FAP) in Wireless Networks and in particular the Radiocoloring problem (RCP) as described in Deliverable D1.1. We study the computational complexity and provide efficient approximation algorithms for the optimizations versions of the radiocoloring problem for some important network topologies (i.e. planar) and kinds of networks (i.e. flat, periodic, hierarchical) as specified in Deliverable D1.1. These results highlight the difficulty of efficiently assigning frequencies in networks that often appear in practice and also provide approximating algorithms that may be applied to get good and simple bandwidth management solutions in reasonable time.

1.1 The Radiocoloring Problem

The Problem of Frequency Assignment in radio networks (FAP) is a well-studied, interesting problem. It is usually modelled by variations of graph coloring. The interference between transmitters are usually modelled by the interference graph G(V,E), where the set V corresponds to the set of transmitters and the set E represents distance constraints. The set of colors represents the available frequencies. In addition, the color of each vertex in a particular assignment gets an integer value which has to satisfy certain inequalities compared to the values of colors of nearby nodes in the interference graph G (frequency- distance constraints). We here study a variation of FAP, called the Radiocoloring Problem (RCP), that models co-channel and adjacent interference constraints.

Deﬁnition 1.1.1 (Radiocoloring) Given a graph G(V,E) consider a function Φ: V → N ∗ such that |Φ(u) − Φ(v)| ≥ 2 if D(u, v) = 1 and |Φ(u) − Φ(v)| ≥ 1 if D(u, v) = 2. The least possible number of colors (order) that can be used to radiocolor G is denoted by Xorder(G). The number ν = maxv∈V Φ(v) − minu∈V Φ(u) + 1 is called span of the radiocoloring of G and the least such number is denoted as Xspan(G).

Real networks usually reserve bandwidth (range of frequencies) rather than distinct frequencies. The objective of an assignment here is to minimize the bandwidth (span) used. The optimization version of RCP related to this objective, called min span RCP,

3 tries to find a radiocoloring for G of minimum span, Xspan(G). However, there are cases where the objective is to minimize the distinct number of frequencies used so that unused frequencies are available for other use by the system. The related optimization version of RCP here, called min order RCP, tries to find a radiocoloring that uses the minimum number of distinct frequencies, Xorder(G). The min span order RCP tries to find one from all minimum span assignments that uses the minimum number of colors. Similarly, the min order span RCP tries to find one from all minimum order assignments that uses the minimum span. Another variation of FAP related to the radiocoloring problem uses the square of a graph G:

Deﬁnition 1.1.2 Given a graph G(V,E), G2 is the graph having the same vertex set V and an edge set E0 : {u, v} ∈ E0 iﬀ D(u, v) ≤ 2 in G.

The related problem tries to color the square of a graph G, G2, with the minimum number of colors, denoted as χ(G2). so that neighbour vertices have diﬀerent colors.

Observation 1.1.1 The min order RCP of G, is equivalent to the problem of vertex 2 2 coloring of G , i.e. Xorder(G) = χ(G ).

This is so because from any coloring of G2 we can get a radiocoloring of G of the same order by doubling the color (integer) assigned to each vertex. On the other hand, every radiocoloring of G is a valid coloring of its square.

1.1.1 Related work

The problem of coloring the square of a graph G has been studied in [45] named as Distance-2-Coloring (D2C). Actually, the D2C problem tries to color the vertices of the graph G so that any two vertices of distance at most two get diﬀerent colors, with the minimum number of colors. Obviously, this problem is equivalent to the problem of coloring the square of the graph G. So, in the following we use the terms coloring the square of the graph, D2C and min order RCP equivalently. In [32] it is proved that the problem of coloring the square of a general graph is NP- complete. In [45] it is proved that the problem remains NP-complete even for the class of planar graphs. In [27] and [22] it has been proved that the problem of min span radiocoloring is NP-complete, even for graphs of diameter 2. Several researchers ([6, 31, 50, 44]) studied the approximability of optimization versions of the radiocoloring problem giving approximations and upper bounds for some interesting families of graphs such outerplanar graphs, k-outerplanar, planar graphs, graphs of bounded treewidth, permutation split, (r,s)-civilized graphs.

4 1.2 Radiocoloring general and planar graphs: Com- plexity and approximations

In [23] we have shown the following: 0 (a) The number of colors Xorder(G) used in the min span order RCP of graph G is diﬀerent from the chromatic number of the square of the graph, X(G2). (b) The radiocoloring problem for general graphs can not be approximated within a factor of n1/2−² by any polynomial algorithm, where n is the network size. However, when restricted to some special cases of graphs, the problem becomes easier. (c) The problems of min span order RCP, min order span and min span RCP are NP-complete for planar graphs. (d) We then present an O(n∆) algorithm that approximates the minimum order of

RCP, Xorder, of a planar graph G by a constant ratio which tends to 2 as the maximum degree ∆ of G increases. (e) Finally, we study the problem of estimating the number of diﬀerent radio colorings of a planar graph G. This is a #P-complete problem (as can be easily seen from our completeness reduction that can be done parsimonious). We employ here techniques of rapidly mixing Markov Chains and the method of coupling for purposes of proving rapid convergence (see e.g. [30]) and we present a fully polynomial randomised approximation scheme (fpras) for estimating the number of radio colorings with λ colors for a planar graph G, when λ ≥ 4∆ + 50. Our algorithm is motivated by a constructive coloring theorem of Heuvel and McGui- ness ([29]). Their construction can lead (as we show) to an O(n2) technique assuming that a planar embedding of G is given. We improve the time complexity of the approximation, and we present a much more simple algorithm to verify and implement. Our algorithm does not need any planar embedding as input. Recently, Agnarsson and Halld´orsson ([3]) presented upper bounds for the chromatic number of square and power graphs (Gk). For planar graphs of large degree (∆ > 749) they obtain an 1.8-approximation while for planar graphs of general degree they get a 2 approximation. Bodlaender et al ([6]) proved recently that the problem of min span radiocoloring, they call it λ-labeling, is NP-complete for bipartite planar graphs, using a reduction which is very similar to our reduction. In the same work the authors presented approximations for the best λ for some interesting families of graphs: outerplanar graphs, graphs of treewidth k, permutation and split graphs. Another relevant work is that of Formann et al ([21]), where the authors proved the 13∆ 2/3 chromatic number of the square of any planar graphs is at most ( 7 ) + Θ(∆ ). Our work has motivated further relevant research. Very recently Moloy and Salavatipour in [39] improved our approximation by providing a constructive upper bound 5 of 3 ∆(G) + constant for a more general problem called L (p,q)-Labeling, which is equivalent to RCP for p = 2 and q = 1.

5 1.2.1 The diﬀerence between min span order radiocoloring and distance-2-coloring a) Planar Graphs

We ﬁrst present (Figure 1.1) an instance of a planar graph G for which the order of min span order RCP of G is diﬀerent from the minimum order of coloring the square of graph G (distance-2-coloring).

Theorem 1.2.1 There is an instance of a planar graph G for which the order of min span order RCP of G is diﬀerent from the minimum order of coloring the square of graph G (distance-2-coloring).

In [23] we prove that for the graph G of Figure 1.1 the order of min span order radiocoloring is 8 while the order of distance-2-coloring is 6.

radiocoloringwithminspanANDorder: coloringthesquareofG,withmin order:

cliquevertex 6 cliquevertex 2 4 1

centralvertex 8 4 2 cliquevertex 3

new1 5 new2 5 vertex 6 7 vertex 3 6

7 1 1 1 3 5 1 8 4 2 3 3 5 2

radial radial vertices vertices order=6(weminimizetheorder) span=8(minimumspan) order=8(minimizeorder)

Figure 1.1: An instance where the problem of min span order radiocoloring and the problem of distance-2-coloring have diﬀerent orders.

b) Arbitrary large diﬀerence between min span order and distance-2-coloring in general graphs

Theorem 1.2.2 For any integer k, there exists an instance of (general) graph G where the order of min span order Radiocoloring is 4k while the minimum order of coloring the square of graph G (distance-2-coloring) is 3k.

In [23] we prove that for the graph G of Figure 1.2 the order of a min order radiocoloring assignment is 3k, while the minimum order distance-2-coloring 4k.

6 cliqueofsizekformedbyvertices level1 oflevel1 cliqueofsize2kformedbyvertices oflevels1,2 cliqueofsizekformedbyvertices level2 oflevel2 cliqueofsize2kformedbyvertices oflevels2,3 level3 cliqueofsizekformedbyvertices oflevel3 cliqueofsize2kformedbyvertices oflevels3,4 level4 cliqueofsizekformedbyvertices oflevel4

Figure 1.2: An instance of a graph of Theorem 1.2.2 where the order of min span radiocoloring and the order of distance-2-coloringlevel1 diﬀers by1,2,3k,4,,5 for,...,k any given k. cliqueofsize2kformedbyvertices

oflevels1,2

+1, +2, +3, +4,...,2 level2 1.2.2 The inapproximabilitycliqueofsize2kformed ofbyverti Radiocoloringces in general net-

oflevels2,3

level3 2 +1,2 +2,2 +3,2 +4,...,3 works cliqueofsize2kformedbyvertices oflevels3,4 The problem of min orderleve Radiocoloringl4 for general1,2, graphs3,4,5,...,k is proved to be hard to be approximated within the ratio of n1/2−² (where n is the number of vertices of the graph), by any polynomial time approximation algorithm. minorder Theorem 1.2.3 The minimum order Radiocoloring Problem for general graphs cannot be approximated within the ratio of n1/2−² unless P=NP.

The reduction is performed by reducing from the ordinary vertex coloring problem which is known to be non-inapproximable within a factor of n1−², see [23].

1.2.3 The NP-Completeness of the RCP for Planar Graphs In [23], we have proved that the decision version of min span radiocoloring remains NP- complete for planar graphs.

Theorem 1.2.4 The min span radiocoloring problem is NP-complete for planar graphs.

The proof is performed by reducing from ordinary 3-coloring of planar graphs. The new graph G0 with the desired properties is obtained by any planar graph G applying local transformation and the component design technique ([26]). We remark that this result is not implied by the NP-completeness result for the distance-2-coloring (which is equivalent to min order RCP), since by Theorem 1.2.1 the two problems are diﬀerent.

1.2.4 A Constant Ratio Approximation Algorithm for Planar Networks The Algorithm

We provide here an approximation algorithm for radiocoloring of planar graphs by mod- ifying the constructive proof of the theorem presented by Heuvel and McGuiness in [29]. The algorithm presented here is of better time complexity (i.e. O(n∆)) compared to the (implicit) algorithm in [29] which needs O(n2). The improvement is achieved by performing the heavy part of the computation of the algorithm only in some instances of G

7 instead of all in [29]. This enables less checking and computations in our algorithm. Also, the behavior of our algorithm is very simple and more time eﬃcient for graphs of small maximum degree. Finally, the algorithm provided here needs no planar embedding of G, as opposed to the algorithm implied in [29]. The main theorem of [29] provides an upper bound for planar graphs for a more general problem called L (p,q)-Labeling, which is equivalent to RCP for p = 2 and q = 1. By setting p = q = 1 and using the observation λ(G; 1, 1) = χ(G2), where χ(G2) is the chromatic number of the graph G2, we get immediately, that:

Theorem 1.2.5 ([29]) If G is a planar graph with maximum degree ∆ then χ(G2) ≤ 2∆ + 25.

The theorem is proved using two Lemmas. Let d(v) the degree of vertex v. The ﬁrst of the two Lemmas is the following:

Lemma 1.2.6 ([29]) Let G be a simple planar graph. Then there exists a vertex v with k neighbors v1, v2, . . . , vk with d(v1) ≤ · · · ≤ d(vk) such that one of the following is true:

(i) k ≤ 2; (ii) k = 3 with d(v1) ≤ 11;

(iii) k = 4 with d(v1) ≤ 7 and d(v2) ≤ 11;

(iv) k = 5 with d(v1) ≤ 6, d(v2) ≤ 7, and d(v3) ≤ 11.

The second Lemma, is quite similar. We apply the following operation to G: For an edge e ∈ E let G/e denote the graph obtained from G by contracting e. We provide below a high level description of our algorithm. Algorithm Radiocoloring(G) [I] Sort the vertices of G by their degree.

[II] If ∆ ≤ 12 then follow Procedure (1) below: Procedure (1): Every planar graph G has at least one vertex of degree ≤ 5. Now, inductively assume that any proper (in vertices) subgraph of G can be radiocolored by 66 colors. Consider a vertex v in G with degree(v) ≤ 5. Delete v from G to get G0. Now recursively radiocolor G0 with 66 colors. The number of colors that v has to avoid is at most 5∆ + 5 ≤ 65. Thus, there is one free color for v.

[III] If ∆ > 12 then

1. Find a vertex v and a neighbor v1 of it, as described in Lemma 1.2.6, and set e = vv1. 2. Form G0 = G/e (G0 = (V 0,E0) with |V 0| = n − 1, while |V | = n) and denote the new vertex in G0 obtained by the contraction of edge e as v0. 0 Modify the sorted degrees of G by deleting v, v1, and inserting v at the appropriate place, and also modify the possible affected degrees of the neighbors of both v and v1. 3. φ(G0) =Radiocoloring(G0)

8 4. Extend φ(G0) to a valid radiocoloring of G: 0 0 (a) Set v1 = v and give to v1 the color of v . (b) Color v with one of the colors used in the radiocoloring φ of G0.

Notice ﬁrst that Procedure (1) produces a proper coloring of G2 with X = 66 colors. Then, assign the frequencies 1, 3,..., 2X − 1 to the obtained color classes of G2. This is a proper radiocoloring of G with the same number of colors.

Theorem 1.2.7 (Performance) The algorithm Radiocoloring(G) outputs a radiocoloring of G using no more than max{66, 2∆ + 25} colors and approximates Xorder(G) by a 25 66 constant factor of at most max{2 + ∆ , ∆ }. It runs in O(n∆) sequential time.

A key observation for the proof of the Theorem is that the graph G0 obtained (G0 = G/e with e is chosen so that Lemma 1.2.6 holds for this edge) has ∆(G0) ≤ ∆(G). Now, the proof is performed by induction. Assume a radiocoloring assignment of G0 using the above number of colors. By the way edge e is chosen (so that to fulﬁlls the requirements of Lemma 1.2.6), we can prove that we can extend this assignment to a radiocoloring assignment of G. See [23] for the proof of the Theorem.

1.2.5 An FPRAS for the Number of Radiocolorings of a Planar Graph Sampling and Counting

Let G be a planar graph of maximum degree ∆ = ∆(G) on vertex set V = {0, 1, . . . , n−1} and C be a set of λ colors. Let φ : V → C be a (proper) radiocoloring of the vertices of G. Such a radiocoloring always exists if λ ≥ 2∆ + 25 and can be found by our O(n∆) time algorithm of the previous section.

Consider the Markov Chain (Xt) whose state space Ω = Ωλ(G) is the set of all radiocolorings of G with λ colors and whose transition probabilities from radiocoloring Xt are modelled by: 1. choose a vertex v ∈ V and a color c ∈ C uniformly at random (u.a.r.) 2. recolor vertex v with color c. If the resulting coloring X0 is a proper radiocoloring 0 then let Xt+1 = X else Xt+1 = Xt. The procedure above is similar to the “Glauber Dynamics” of an antiferromagnetic Potts model at zero temperature, and was used by [30] to estimate the number of proper colorings of any low degree graph with k colors.

The Markov Chain (Xt), which we refer to in the sequel as M(G, λ), is ergodic, provided λ ≥ 2∆ + 26, in which case its stationary distribution is uniform over Ω. We show here that M(G, λ) is rapidly mixing i.e. converges, in time polynomial in n, to a close approximation of the stationary distribution, provided that λ ≥ 2(2∆ + 25). This can be used to get a fully polynomial randomised approximation scheme (fpras) for the number of radiocolorings of a planar graph G with λ colors, in the case where λ ≥ 4∆ + 50. For some deﬁnitions and measures used below see [23].

Theorem 1.2.8 The above method leads to a fully polynomial randomized approximation scheme for the number of radiocolorings of a planar graph G with λ colors, provided that λ > 2(2∆ + 25), where ∆ is the maximum degree of G.

9 1.3 Eﬃcient coloring of Squares of Planar Graphs

In [2] we have considered the min order RCP problem on planar graphs. The motivation for this work is important since most real networks are planar. We provide a new coloring algorithm, called MDsatur, and we analyze its performance on the square of any planar graph. As we have already said, in section 1.1, this problem is equivalent to the min order RCP. We prove that MDsatur colors the square of any planar graph G with at most 1.5∆(G) + c colors, where c is a constant, when ∆(G) ≥ 8. When ∆(G) < 8 then as concluded by the work of Heuvel and McGuiness [29] a constant number of colors are enough to color the squares of such planar graphs. Hence, we do not analyze the performance of our algorithm on such graphs. MDsatur is inspired by algorithm Dsatur of Br´elaz [7], which is a greedy coloring algorithm with very good behaviour on classes of graphs with similar characteristics as planar graphs and their squares. Dsatur colors the vertices of a given graph, say G, one at a time based on a dynamic order, O, of them. Dsatur ﬁnds at each point, i.e. i, which of the currently uncolored vertices of G will be the ith element of O (denoted by Oi). Then, it correctly colors Oi with the smallest available color (i.e. it is not assigned to any of its colored neighbors). If Oi = v, then v is the current uncolored vertex of G which has the biggest value on its degree of saturation.

Deﬁnition 1.3.1 The degree of saturation of vertex v is denoted by ds(v) and is the number of distinct colors that have been assigned to its neighbors.

Our algorithm, MDsatur, also colors the vertices of a given graph, i.e. G2, in the same way (namely, greedily choosing the smallest available color for each vertex) but following a diﬀerent dynamic order on its vertices. Our algorithm diﬀers from the original algorithm Dsatur in the way of growing the dynamic order O. More precisely, at each of its points, i.e. i, it orders the current uncolored vertices of G by taking into account not only their degree of saturation, but also by considering how ‘close’ is each of the potential vertices for the position i of order O to the current last colored vertex, let us call it CurrLast, following a procedure similar to Breadth First Search. The term ‘close’ is related to the closeness of the behaviour of the neighbors of a vertex, i.e. u, to its behaviour. The most ‘close’ neighbor of u is the one whose coloring is determined at most by the colors of the other neighbors of u. In most of the cases the location of this neighbor of u, in an embedding of G in the plane, is very near to the location of u.

1.3.1 Presentation of the coloring Algorithm M Dsatur

In this paragraph, we present the coloring algorithm MDsatur in pseudo-code.

10 Algorithm: MDsatur Input : a graph G = (V,E). Output: a coloring of G. Begin

1. (a) Set each vertex of G as uncolored, and set a dynamic order on them, called O, as empty. (b) Select any vertex of G.

Add it in the beginning of O (i.e. in O1), and color it with color 1.

2. For i = 2 to n = |V | do

(a) Set X to be the set of the current uncolored vertices which have the maximum value on their degree of saturation. (b) For each vertex v ∈ X do Find which colored neighbor of v is colored ﬁrst (i.e. its oldest colored neighbor) (see req. A2). (c) Select from X the vertex v which has the oldest colored neighbor, denoted by ON, and also has the maximum number of common neighbors with the last colored vertex (namely Oi−1) (see req. A3).

Set Oi = v. If more than one of the vertices of X satisﬁes these requirements, then select at random one of them with equal probability. If none of the uncolored neighbors of ON, which are in X, has common

neighbors with vertex Oi−1, then select one of them at random. (d) color v with the smallest of its allowable colors. If none of the used colors is allowable for this vertex, insert a new color.

3. Return the color of each vertex.

End

This algorithm consists a novel approach in the coloring of squares of planar graphs. The inspiration for the design of MDsatur originates from the behaviour of the coloring algorithms Dsatur and NoChoice of Br´elaz and Turner, respectively, on random [47] and k-colorable random graphs [48], respectively. These algorithms have very good and optimal performance on the above families of graphs. This happens, because each vertex of these graphs belongs to a clique of size almost maximum and maximum, respectively, and the chromatic number of these graphs is strongly related to the size of their maximum clique. For our case, it is easily concluded from the known results, that the chromatic number of the square of any planar graph G is strongly related to the size of its maximum clique.

11 More precisely it is known by Molloy and Salavatipour that χ(G2) ≤ 1.66∆ + 24 see [39]. Hence, χ(G2) ≤ 1.66ω(G2) + 24. An experimental evaluation of the coloring algorithm Dsatur on squares of planar graphs reveals some interesting properties for the performance of this algorithm on this class of graphs. Focusing our attention on the structure of the planar graphs where Dsatur has its worst performance and studying carefully its behaviour we conclude, that we could guide it to have better performance. Based on these experimental results we modiﬁed Dsatur, in a not straightforward way, by taking into account additional requirements. These limit the number of the potential uncolored vertices of G2 which claim to be the 2 ith element of order O (Oi). Suppose that we apply MDsatur to graph G . Let i be the current point of this algorithm in its application on G2. Then, the current uncolored vertex which manages to become the vertex in position i of O, let it be v, satisﬁes the following requirements in turn: A1) has the maximum value on its degree of saturation in the current point of this algorithm (as Dsatur), A2) has the oldest colored neighbor (i.e. sits in the leftmost position in O among the neighbors of v) and A3) is the ‘closer’ neighbor of CurrLast. Our algorithm has better performance than Dsatur, because it locally colors a given graph and the colored subgraph obtained in each point of its application on this graph is more dense, than the corresponding graph obtained by Dsatur.

1.3.2 Our results Our main result is presented in the following Theorem.

Theorem 1.3.1 Let G be a planar graph. Algorithm MDsatur colors G2 using at most 1.5∆(G) + c colors, where c is a constant, when ∆(G) ≥ 8.

In order to prove the assertion of this Theorem we have characterized the worst case graph instances of squares of planar graphs for our algorithm according to the next The- orem. Then, by a case analysis we prove that MDsatur colours any such graph with at most 1.5∆(G) + c colors.

Theorem 1.3.2 The minimal worst case squares of planar graphs for the coloring algorithm MDsatur are the squares of the graphs obtained by any planar r−regular graph, replacing each of its edges, i.e. e, with a set of vertices and joining them with e0s end- points.

The proof of Theorem 1.3.2 is based on the fact, that MDsatur is a greedy coloring algorithm (it colors the vertices of a given graph based on an ordering of its vertices with an available already used color. If any such color does not exist, then it uses a new one.) without backtracking. Suppose that MDsatur is applied on a graph G and let H be an uncolored subgraph of G in an intermediate point of the algorithm. If the colored neighbors of each vertex of H have all the currently used colors (used in the colouring of H0), then MDsatur has to increase the number of colors used at least by χ(H). This process can continue recursively on the current uncolouled subgraph of G. We denote the graph H in the ﬁrst case by H1. After the coloring of H1, the value of XMDsatur (that denotes the number of colors already used) is increased and it continues to increase whenever there is another uncolored subgraph of G such that all the vertices of this graph have colored neighbors with XMDsatur diﬀerent colors. Seeing abstractly this procedure we can notice that there is a sequence of graphs having the properties of graph H1. We

12 denote the graphs of this sequence as Hi, for i ≥ 1. Observe that the coloring of graph Hi is based on the coloring of each of the graphs Hj, for j < i. The index of each graph Hi shows its level in the above sequence. In the case of squares of planar graphs, we prove that the maximum number of levels of graph H is three. Thus, we focus our attention on the structure of graphs H0,H1,H2. To specify the possible structures of these graphs we have ﬁrstly determined the structure of each possible square root of any clique of size more than eight.

1.3.3 Conclusions Our algorithm colors the square of any planar graph G with at most 1.5∆(G) + c colors. We conjecture that this is not a tight bound in the case where χ(G2) = ∆(G) + 1. So, this is an open problem for this case. We also conjecture that MDsatur colors any planar graph with at most 5 colors.

1.4 Radiocoloring Periodic Planar Graphs: PSPACE-completeness and appoximations

In [25] we investigate the min span RCP for an interesting family of infinite planar graphs, called periodic planar graphs. A periodic graph G is defined by an infinite sequence of repetitions of the same finite graph Gi(Vi,Ei). The edge set of G is derived by connecting some of the vertices of each iteration Gi to some of the vertices of the next iteration Gi+1, the same for all iterations. Infinite periodic graphs usually represent finite networks that accept periodic (in time, e.g. daily) requests for frequency assignment. We note that periodic interference graphs usually represent networks of great practical interest, since in many networks the requests for frequency assignment exhibit some periodic behavior. That is, the network accepts periodic (e.g. daily) requests for frequency assignment. Each request has a starting and ending time and a node where it is applied. Two requests interfere if they apply for nearby nodes and their time intervals overlap. The assignment should be such that there is no time overlap between any two nearby requests of the same or the preceding and following periods of requests. Alternatively, infinite periodic graphs can model very large networks produced by the repetition of a small graph. Note in this context that many real networks consist of the repetition of the same component. We focus here on planar periodic graphs, because in many cases real networks are planar and because of the independent mathematical interest of this family of graphs.

Definition 1.4.1 Linear Periodic Planar Graph G: A linear periodic planar graph is defined as follows: Let Ge be an arbitrary finite connected planar graph. Let V the vertex set of Ge. Let e also E0 be the edge set of G. Let E+ be a specific set of ordered pairs (u, v) of the nodes e of G. Note that E+ must be a set of ordered pairs of vertices whose connection according to the rule (c2) below leads to planarity preservation. Consider the two-way infinite sequence of graphs ...,Gi,Gi+1,..., where each Gi is isomorphic to Ge. The infinite graph G is obtained from this sequence as follows:

(a) We assume a line (in fact, any 1-dimensional inﬁnite simple curve) on which we select discrete points . . . , i, i + 1, i + 2,..., such that:

13 (a1) Each point in the line is replaced by Ge.

(a2) Each edge (i, i + 1) in the line is replaced by E+. (a3) For any ﬁnite subset of consecutive points in the line, replacing the points e of the line by graphs G end the edges between them by E+, the resulting graph is planar.

(b) The vertex set of G is the union of the vertex sets of the sequence ...,Gi,Gi+1,....

(c) The edges of G are (c1) The union of edge sets of the sequence of Gis (i.e., the e e edge set E0 of G) (c2) For each pair of adjacent copies of G, call them Gi, Gi+1, we use the E+ specification of G to connect the nodes of Gi corresponding to the first elements of the pairs in E+ to the nodes of Gi+1, corresponding to the second elements of the pairs in E+. e We denote a linear periodic planar graph by G = (G(V,E0),E+). This pair is called the finite specification of G.

Summary of Results In [25] we have given the following results: 1. We first prove that the min span RCP is PSPACE-complete for periodic planar graphs. (The space is polynomial with respect to the size of the finite specification of G.)

2. We provide an O(n(∆(G)) time algorithm, which obtains a radiocoloring of a periodic planar graph G that approximates the minimum span within a ratio which tends to R as ∆(G) tends to inﬁnity. The term R is the approximation ratio obtained by any approximation algorithm for the min span RCP of a ﬁnite planar graph G achieving a span of at most R · ∆(G) + constant. The best known radiocoloring algorithm for planar graphs has an approximation ratio which tends to 1, 66 [39], so this is the approximation ratio obtained by our algorithm too.

Related Work A model for periodic graphs (called l-dimensional periodic graphs) was first presented by Orlin in [41]. The model of periodic graphs considered in this work is similar to that of Orlin for the 1-dimensional case, l = 1 (also called 1-dimensional periodically specified graphs or simply periodically specified graphs), when restricted to planar instances. The complexity of various basic problems of periodically specified graphs was studied by Orlin [41] and Wanke [49]. In [41, 36, 43] it is proved that the problems of Maximum Independent Set (MIS), Hamiltonial Path, Partition into Triangles, SAT, 3-coloring for periodically specified graphs are PSPACE-complete. The appoximability of basic problems on infinite periodic graphs was studied by several researchers ([19, 28, 42]) giving efficient algorithms for solving problems such as determining strongly connected components, testing the existence of cycles, bipartiteness, planarity and minimum cost spanning forests for periodically specified graphs. Marathe et al [35] presented several PSPACE-hardness results and also efficient approximation schemes for partitioning problems including MIS, min vertex cover and max- SAT for periodically specified graphs when restricted to planar instances. However, their approximation technique for periodically specified graphs (illustrated for the MIS problem) can not directly apply for coloring problems, considered here, because it takes the

14 union of partial solutions-subsets of the inﬁnite graph and thus it does not consider all the vertices; something not allowed in coloring problems.

1.4.1 Embeddings of Periodic Planar Graphs Studying periodic planar graphs, we use the notions of embedding and discover some interesting properties on the embeddings of a linear periodic planar graph. We use the notion of an embedding of a planar graph. Deﬁnition 1.4.2 Planar Embedding (of a periodic graph G)([40]): For each node v of G, there is an adjacency list, such that all neighbours of v appear in clockwise order with respect to an actual drawing of G. The following Lemma reveals important information about the structure of a linear periodic planar graph. Lemma 1.4.1 Any linear periodic planar graph G can be embedded in the plane by interchanging at most two diﬀerent planar embeddings of the graph obtained by an iteration i e of G, Gi (which is isomorphic to G) and the set of edges connecting Gi with the previous e and next iterations, sets Ei−, Ei+ (each of which is equal to E+), called Extended G. The Lemma is proved basically via exhaustive check of all possible edge sets that can lead to a planar graph.

1.4.2 The PSPACE-Completeness of min span RCP for Periodic Planar Graphs We have proved that min span RCP is PSPACE-complete for periodic planar graphs. In order to show this, we need to prove that a number of problems are PSPACE-complete. Our PSPACE-completeness proofs utilize known constructions for the NP-completeness of corresponding problems for ordinary graphs. However, note that applying those constructions on the infinite periodic graph G we need infinite time to get the transformed graph G0 with the desired properties. We manage to apply the transformation only to a part of the infinite graph (an iteration) and from the obtained graph to get the transformation of the whole infinite periodic graph, thus the new graph G0, in time polynomial to the size of the finite specification of G. Moreover, the new graph fulfills the desired properties. These are achieved by exploiting some ‘locality characteristics’ that the constructions utilized here exhibit, i.e. the construction applies locally on a part (vertex or edge) of this part involving only information of the neighborhood of the part and it affects only to this neighborhood. This, combined with the repetitive structure of an infinite periodic graph enables us to get constructions of polynomial time in the size of the finite specification of the graph. The main complexity Theorem of [25] is the following: Theorem 1.4.2 Let r ≥ 8 be an even integer. The problem of deciding whether a periodic e planar graph G = (G(V,E0),E+) of maximal degree r −2 can be radiocolored using a span of size at most r is PSPACE-complete. The reduction adapts the ‘local’ construction by Bodlaender et al in [6] to show NP- completeness of min span RCP of ordinary planar graphs reducing it from the 3-coloring of planar graphs with a given 4-edge coloring. Here we reduce from 3-coloring of periodic planar graphs with a given 4-edge coloring which we also prove to be PSPACE-complete using the same methodology. For the proof of the Theorem see [25].

15 1.4.3 An Eﬃcient, Constant Ratio Approximation Algorithm for min span RCP for Periodic Planar Graphs We present an eﬃcient time, constant ratio approximation algorithm that approximates the min span radiocoloring problem for periodic planar graphs with the same ratio as the ratio obtained by the best known approximation algorithm for ordinary planar graphs

(which we use as a subroutine for the ﬁnite speciﬁcation), for the same problem.

The modiﬁed graph 3=2

1=1

The algorithm groups together every four successive iterations of the infinite graph. Call 2=2 the j − th such group as Ggroup j. Denote the first graph of such a group as G(j)1 or G1, the second as G(j)2 or G2 and so on until the fourth. Consider any such group Ggroup j. The algorithm modifies the graph to obtain a new 0 graph, Ggroup j as follows: The new graph has the same vertex and edge set as the graph Ggroup j except from the following modifications on the first and the fourth graphs of group Ggroup j: The edges of the first graph of the group connecting it to the previous iteration are removed. Also, the edges of the last iteration, the fourth, connecting a vertex of this iteration to the next iteration are removed. Then for each removed edge uv u ∈ G(j)1 and v ∈ G , add edge vu0 where u0 is the corresponding to u vertex in G . An example (j−1)4 G (j)4 groupj G of the graph obtained by a periodic graph is illustrated in Figure 1.3.group(j+1)

added

u''≠ G j*3-1 u d u ele ' ted deleted u ≠G G G j*3+1 1 G2 3 G4

E1 E2 E1 E2

0 Figure 1.3: The graph Ggroup j produced by the Group j of the periodic graph G

0 The graph Ggroup j has two critical properties compared to the initial periodic planar 0 graph G: (i) it has the same maximum degree as the initial graph G, i.e. ∆(Ggroup j) = ∆(G) and (ii) using Lemma 1.4.1 it is proved that it is a planar graph too.

The Periodic Radiocoloring Partitioning Algorithm (PRPA) The following deﬁnition is needed by the Algorithm. The deﬁnition uses the observation that the optimal span, S∗, of a radiocoloring of a graph G with maximum degree ∆(G) is clearly S∗ ≥ ∆(G).

Deﬁnition 1.4.3 RC Algorithm: Let an RC Algorithm be any known min span radiocoloring polynomial time approximation algorithm for ﬁnite planar graphs with performance ratio R (when ∆(G) is used as a lower bound), i.e. if S∗ is the optimal span and SRC is the span obtained by the algorithm then there are constants R > 1 and b such that ∗ ∆(G) ≤ S ≤ SRC ≤ R · ∆(G) + b For example the algorithm of [29] is an RC algorithm with R = 2 and b = 35.

16 Algorithm PRPA

0 1. Run an RC algorithm, on graph Ggroup j. Let SRC be the span obtained 0 by RC on Ggroup j.

2. For all j = 1, 2,... color the four graphs G(j−1)∗4+1, G(j−1)∗4+2, G(j−1)∗4+3, , G(j−1)∗4+4 of the group Ggroup j as follows: Set the color of each vertex of graph G(j−1)∗4+k, k = 1, 2, 3, 4 to the color of 0 its corresponding vertex, in Vk of V (Ggroup j). Step 2 produces a radiocoloring of the whole periodic graph G with span SRC .

Theorem 1.4.3 The Algorithm PRPA produces a radiocoloring on the inﬁnite linear periodic graph G, runs in time O(T (RC)) and approximates the span within an asymptotic 0 ratio of R, where T (RC) is the time needed for the RC Algorithm to run on Ggroup j.

Since the modiﬁed graph is planar and has the same maximum degree as the inﬁnity graph, we get:

Corollary 1.4.4 If the RC algorithm is that of [39] for planar graphs, then algorithm PRPA has R = 1, 66 and b = 24 and runs in O(n(∆(G)) time, where n = |Vi|.

1.5 Radiocoloring Hierarchically Speciﬁed Planar Graphs: PSPACE-completeness and appoximations

Many practical applications of graph theory and combinatorial optimization in CAD systems, VLSI design, parallel programming and software engineering involve the processing of large (but regular) objects constructed in a systematic manner from smaller and more manageable components. As a result, the graphs that abstract such circuits (designs) also have a regular structure and are defined in a systematic manner using smaller graphs. The methods for specifying such large but regular objects by small specifications are referred to as succinct specifications. One way to succinctly represent objects is to specify the graph hierarchically. Hierarchical specifications are more concise in describing objects than ordinary graph representations. A well known hierarchical specification model, considered in this work, is that of Lengauer, introduced in [33, 34], referred to as L-specifications. Real communication networks, especially wireless and large ones, that may be struc- tured in a hierarchical way and are usually planar. In [1] we study min order RCP on L-specified hierarchical graphs.

1.5.1 Summary of Results In [1] we investigate the computational complexity and provide eﬃcient approximation algorithms for the RCP on a class of L-speciﬁed hierarchical planar graphs which we call Well-Separated (WS) graphs. In such graphs, levels in the hierarchy are allowed to directly connect only to their immediate descendants. In particular:

1. We prove that the decision version of the RCP for Well-Separated L-speciﬁed hierarchical planar graphs is PSPACE-complete.

17 2. We present two approximation algorithms for RCP for this class of graphs. These algorithms offer alternative trade-offs between the quality and the efficiency of the solution achieved. The first one is a simple and very efficient 3,33-approximation algorithm, while the second one achieves a better solution; it is a 2,66-approximation algorithm, but is less efficient, although of polynomial time.

We note that the class of WS L-specified hierarchical graphs considered here can lead to graphs that are exponentially large in the size of their specification. The WS class is a subclass of the class of L-specified hierarchical graphs considered in [35], called k-level-restricted graphs.

1.5.2 Related Work In a fundamental work, Lengauer and Wagner [34] proved that the following problems are PSPACE-complete for L-specified hierarchical graphs: 3-coloring, hamiltonian circuit and path, monotone circuit value, network flow and independent set. For L-specified graphs, Lengauer ([33]) have given efficient algorithms to solve several important graph theoretic problems including 2-coloring, min spanning forest and planarity testing. Marathe et al in [36, 35] studied the complexity and provided approximation schemes for several graph theoretic problems for L-specified hierarchical planar graphs including maximum independent set, minimum vertex cover, minimum edge dominating set, max 3SAT and max cut. We remark that the PSPACE-completeness proof of planar 3-COLORING of WS L-specified hierarchical graphs provided in this work, is not implied by known PSPACE- completeness results of the same problem for similar (but different) classes of planar graphs. This is so because our PSPACE-completeness proof for planar 3-COLORING concerns a subclass studied in [36] of the L-specified hierarchical planar graphs for the same problem. Moreover, the PSPACE-completeness proof of planar 3-COLORING of [37] for L-specified hierarchical graphs which are simultaneously planar and unit disks concerns a different class of hierarchical planar graphs than the class of WS L-specified hierarchical planar graphs considered here. Note, also that most of the work done so far on approximations of PSPACE-complete problems, has basically addressed such ‘finding a subset’ problems and not coloring problems. The methodologies applied for such problems, such as maximum independent set (MIS) in [35], do not directly apply to coloring problems, since they exclude from a solution some vertices of each graph of the L-specification of the hierarchical graph, something not allowed in graph coloring problems. To our knowledge, the only work studying approximations to coloring problems on hierarchical graphs is the work of [37]. We remark that the best currently known approximation ratio for the RCP on ordinary (non-hierarchical) planar graphs (which are much simpler to color than the hierarchical ones) is 1,66 ([39]). Also, the only known results on any kind of coloring problems have been shown for the vertex coloring for a special kind of hierarchical graphs (k-level restricted unit disk graphs) achieving a 6-approximation solution ([37]). We do not see an easy way of using their algorithm for the radiocoloring problem studied here.

1.5.3 Preliminaries We study the RCP on hierarchical graphs as speciﬁed by Lengauer [33].

18 Definition 1.5.1 (L-specifications, [33]) An L-specification Γ = (G1, ··· ,Gi, ··· ,Gn), where n is the number of levels in the specification, of a graph G is a sequence of labeled undirected simple graphs Gi called cells. The graph Gi has mi edges and ni vertices. The pi of the vertices are called pins. The other (ni −pi) vertices are called inner vertices. The ri of the inner vertices are called nonterminals. The (ni −ri) vertices are called terminals. The remaining ni − pi − ri vertices of Gi that are neither pins nor nonterminals are called explicit vertices. Each pin of Gi has a unique label, its name. The pins are assumed to be numbered from 1 to pi. Each nonterminal in Gi has two labels (v, t), a name and a type. The type t of a nonterminal in Gi is a symbol from G1, ··· ,Gi−1. The neighbours of a nonterminal vertex must be terminals. If a nonterminal vertex v is of type Gj in Gi, 1 ≤ j ≤ i − 1, then v has degree pj and each terminal vertex that is a neighbor of v has a distinct label (v, l) such that 1 ≤ l ≤ pj. We say that the neighbor of v labeled (v, l) matches the l-th pin of Gj.

:explicitvertices :pins :nonterminals G3

G G 1 1 G 2 G G1 G2 1 HierarchyTreeofG

G3 G2

G G2 G1 G1 2 G1

G G3 G1 1 (a) (b)

Figure 1.4: (a) An L-speciﬁcation Γ = (G1,G2,G3) of a graph G and its hierarchy tree HT (G). (b) The expansion of the graph G.

See Figure 1.4(a) for an example of an L-specification. Note that a terminal vertex may be a neighbor of several nonterminal vertices. For simplicity reasons we assumed 2 that n = maxi{|Gi|}. Thus, the size of Γ, is O(n ). Definition 1.5.2 (Expansion of an L-specificied hierarchical graph, [33]) Let any L-specified hierarchical graph, given by Γ = (G1, ··· ,Gn). The expanded graph E(Γ) (i.e. the graph associated with Γ) is iteratively obtained as follows: k = 1: E(Γ) = G1. k > 1: Repeat the following step for each nonterminal v of Gk say of the type Gj: delete v and the edges incident on v. Insert a copy of E(Γj) by identifying the l-th pin of E(Γj) with the node in Gk that is labeled (v, l). The inserted copy of E(Γj) is called a subcell of Gk. For example, the expansion of the hierarchical graph G of Fig. 1.4(a) is shown in Fig. 1.4(b).We consider hierarchical planar graphs as studied in [33]: Definition 1.5.3 (Strongly planar hierarchical graph, [33]) An L-specified hierarchical graph G given by Γ = (G1,...,Gn) is strongly planar if E(Γ) has a planar embedding such that for each E(Γi) all pins of it occur around a common face and the rest of E(Γi) is completely inside this face.

19 In fact, we here study a subclass of strongly planar hierarchical graphs, where additionally to the above condition, all graphs Gi, 1 ≤ i ≤ n, are planar. We call this class as fully planar hierarchical graphs. In the sequel, and when there is no ambiguity, we refer to such graphs simply as hierarchical planar graphs. Moreover, we concentrate on a class of L-specified hierarchical graphs which we call Well-Separated (WS) graphs, defined in the sequel using the followings: Consider an L-specified hierarchical graph G, given by Γ = (G1,...,Gn). For each graph Gi (1 ≤ i ≤ n), we define the following subgraphs:

Deﬁnition 1.5.4 Inner subgraph of graph Gi, Gin i: it is induced by the explicit vertices of Gi not connected to any pin or nonterminal of Gi.

Deﬁnition 1.5.5 Outer-up subgraph of graph Gi, GoutUp i: it is induced by the explicit vertices of Gi connected to at least one pin of Gi.

Deﬁnition 1.5.6 Outer-down subgraph of graph Gi, GoutDown i: it is induced by the explicit vertices of Gi connected to at least one nonterminal of Gi.

Remark. 1.5.1 Generally, an explicit vertex of Gi might belong to both outer-up, outer- down subgraphs of Gi. We have studied the following class of graphs:

Deﬁnition 1.5.7 (Well-Separated, WS) We call Well-Separated graphs the class of L-speciﬁed hierarchical graphs of which any explicit vertex of Gi, 1 ≤ i ≤ n, belongs either to GoutDown i or GoutUp i or none of them, but not to both of them. Moreover, any vertex of GoutDown i is located at distance at least 3 from any vertex of GoutUp i.

Observe that the WS class of hierarchical graphs is testable in time polynomial in the size of the L-speciﬁcation of a hierarchical graph. Note also that the WS class is a subclass of k-level-restricted graphs, which is another class of L-speciﬁed hierarchical graphs, studied by Marathe et al. [35] (see [1] for a proof).

Deﬁnition 1.5.8 The maximum degree of a hierarchical graph G, ∆(G), is the maximum degree of a vertex in the expansion of the graph, E(Γ).

The following result is needed by the approximation algorithms presented.

Theorem 1.5.1 [50] Let G(V,E) be a k-tree of n vertices given by its tree-decomposition, let C be a set of colors, and let α = |C|. Then, it can be determined in polynomial time T (n, k), whether G has a radiocoloring that uses the colors of set C, and if such a radiocoloring exists, it can found in the same time, where T (n, k) = O(n(2α +1)22(k+1)(l+2)+1 +n3), l = 2 and n = |V |.

1.5.4 The Complexity of the Radiocoloring Problem In [1], we study the complexity of RCP on L-speciﬁed hierarchical planar graphs. A critical observation about the constructions utilized in the PSPACE-completeness proofs, is that 0 the transformations are local ([26]). I.e, given any hierarchical graph G, the graph Gi obtained from each Gi, is the same for all appearances of Gi in the hierarchy tree of G. Thus, the resulting hierarchical graph G0 can be computed in time polynomial in the size of the L-speciﬁcation of the graph G.

20 Another important issue for the PSPACE-completeness reductions is whether an already known NP-completeness proof for the same problem, that fulfills such locality characteristics, can be modified so that to apply for a hierarchical graph G. This technique has been used in previous papers to get PSPACE-completeness results for a number of problems considered (e.g. [37]). In our case, there was no such ‘local’ NP-completeness reduction available. The corresponding NP-completeness reductions that both could be adapted for the hierarchical case are the reductions of [45, 38]. However, although the reduction of [45] is local that of [38] is not. Henceforth, they can not be used to get the PSPACE-completeness of L-specified hierarchical planar graphs. For these reasons, we have developed a new NP-completeness proof for the RCP of ordinary planar graphs which reduces it from the problem of 3-coloring planar graphs. The construction satisfies the desired locality characteristics and thus, we can utilize it to get the PSPACE-completeness proof of the RCP for L-specified hierarchical planar graphs.

1.5.5 The NP-completeness of RCP for planar graphs We have provided a new NP-completeness proof for the problem of radiocoloring for ordinary (non-hierarchical) planar graphs, which is ‘local’. We remark that this reduction is the only one that works for the cases where ∆(G) < 7 (∆(G) ≥ 3), in contrast to the only known NP-completeness proof of [45].

Theorem 1.5.2 The following decision problem is NP-complete: Input: A planar graph G(V,E). Question: Is λ(G) ≤ 4?

1.5.6 The PSPACE-completeness of RCP for Hierarchical Pla- nar Graphs The PSPACE-completeness reduction for the RCP of L-speciﬁed hierarchical planar graphs, utilizes the ‘local’ construction of the NP-completeness proof of the radiocoloring problem for ordinary planar graphs, given in section 1.5.5, Theorem 1.5.2, in order to be of polynomial time to the size of the L-speciﬁcation.

Theorem 1.5.3 The following decision problem is PSPACE-complete: Input: A WS fully planar hierarchical graph G, given by the L-speciﬁcation Γ = (G1,...,Gn). Question: Is λ(G) ≤ 4?

1.5.7 Approximations to RCP for WS Fully Planar Graphs In this section we present two approximation algorithms for the min order RCP on WS fully planar hierarchical graphs: a simple and fast algorithm, that achieves an approximation ratio of 3,33, and a more sophisticated one which, being still polynomial, achieves a 2,66-approximation ratio. These algorithms offer alternative options that trade-off the efficiency of the algorithm and the quality of the solution achieved. Both algorithms utilize a bottom up methodology of radiocoloring an L-specified hierarchical planar graph G, given by Γ = (G1,...,Gn). Actually, they compute at most n − i radiocolorings for a

21 subgraph of each graph Gi, 1 ≤ i ≤ n, and use these radiocolorings for all copies of Gi in the expansion of G, E(Γ). This enables them to run in time only polynomial to the size of the L-specification of G. More analytically, we wish to compute only one radiocoloring assignment for each Gi, and use this in all appearances of Gi in the expansion of G, E(Γ). However, due to the structure of L-specified hierarchical graphs, the distance two neighborhood of the outer vertices of each Gi, may differentiate for every call of Gi by other graphs Gj. Henceforth, a radiocoloring for such a vertex (the outer ones) may becomes invalid due to a change of the distance two neighborhood of the vertex. Since each graph Gi may be called by at most n − i other graphs, we need to compute at most n − i radiocolorings of those (outer) vertices. Moreover, we need to guarantee that the different radiocolorings of the outer part of Gi do not introduce any implication in the radiocoloring of its inner part. By having only one radiocoloring for the inner part of Gi, we manage to have also no implications to the radiocoloring of the subtree of Gi, HT (Gi). Based on this design approach, both algorithms partition appropriately, each Gi into three parts: (1) the inner part, (2) the outer up and (2) the outer down part. Remark, that these subgraphs might be different from the inner, outer up, outer down subgraphs of Gi defined in the Definitions 1.5.4, 1.5.5 and 1.5.6. Then, the algorithms radiocolor the inner part of Gi only once, using, each of them, a different method. Both of them, they group and radiocolor the outer down part of it together with the outer up parts of the graphs called by it, using the best known radiocoloring algorithm which is a 1, 66-approximation.

A 3,33-approximation Algorithm HRC1

We first provide a simple and efficient algorithm (HRC1) that achieves a 3,33- approximation for RCP on WS fully planar hierarchical graphs. Let A1,B1 two disjoint 5 sets of colors of size 3 ∆(G) + 24 each, where ∆(G) is the maximum degree of G. Overview of the HRC1 Algorithm: First, the algorithm defines for each Gi its inner, outer up and outer down parts to be the inner, outer up and outer down subgraphs Gin i, GoutUp i and GoutDown i, respectively. Then, it radiocolors the inner part of the graph Gi using a known 1, 66-approximation algorithm [39]) using the color set A1. Also, it radiocolors the outer down part of the graph Gi together with the outer up parts of its children using the 1,66-approximation algorithm with the color set B1.

Theorem 1.5.4 Algorithm HRC1(G) produces a radiocoloring of a WS fully planar hierarchical graph G in time O(n5) and achieves a 3,33-approximation ratio.

For, a detailed description of the HRC1 algorithm and the proof of Theorem 1.5.4, see [1].

A 2,66-approximation Algorithm HRC2 Overview of the Algorithm: We provide a more sophisticated radiocoloring algorithm, that achieves a 2,66-approximation ratio rather than 3,33, for fully planar hierarchical graphs of class WS. The basic idea of the algorithm, called HRC2, is to partition the vertices of each graph Gi into outerplanar levels using a BFS (similar to [5, 31]) and deﬁne the three parts of each Gi based on this search.

22 The outer up part of Gi consists of the first level of the BFS tree obtained. Thus, it is the outer up subgraph of Gi, GoutUp i. The outer down part of Gi consists of the graph induced by the vertices of the BFS tree of levels D up to the end of the tree, where D is the first level of the tree having an outer down vertex of Gi. The inner part of Gi is the rest of the graph Gi. More analytically, the inner part of Gi is radiocolored as follows: Radiocolor every two successive levels of the inner part of Gi optimally interchanging color sets A, B, where |A| = |B| = OPT (G) and OPT (G) is the optimal number of colors needed to radiocolor G. This can be achieved in polynomial time and without any conflicts as we prove: We first show that any two successive levels, call them a double level, is a 4-outerplanar graph. Thus, as proved in [4], it is a bounded treewidth graph. Consequently, applying Theorem 1.5.1, each double level can be radiocolored optimally in polynomial time. Moreover, by the BFS partitioning procedure, there is no conflict between double levels colored using the same color set. The outer down part of Gi is radiocolored together with the outer up parts of its children using a known (2-approximation) radiocoloring algorithm for ordinary planar graphs using color sets A and C, where |C| = |A| = OPT (G). Since the algorithm uses only color sets A, B, C, it has a 2,66-approximation algorithm.

Theorem 1.5.5 HRC2 Algorithm produces a valid radiocoloring of a WS fully planar hierarchical graph G using at most 2, 66 OPT (G) + 48 colors, achieving a 2,66 approximation ratio. It runs in O(n2 · T (n, k) + n5) time, where T (n, k) is a polynomial time function for the optimal radiocoloring of a k-tree of size n, speciﬁed in [50] (k = 11), (see Theorem 1.5.1).

For, a detailed description of the HRC2 algorithm and the proof of Theorem 1.5.5, see [1].

23 24 Chapter 2

Fault Tolerant Broadcasting, Energy Consumption and Connectivity of WNs

Research activity has focused on two main topics, both related to (Ad-Hoc) Multi-Hop Wireless Networks (in the following, shortly denoted as WN):

1. Fault Tolerant protocols, and

2. Relationship between energy consumption and connectivity properties.

Such topics have received a large attention from the Computer Science Community in recent years. In what follows, we shall describe our most relevant achievements.

2.1 Fault Tolerant Broadcasting in WN

One major communication operation in WNs is broadcasting, which consists in transmit- ting a single message originated at a given source s to all the other nodes in the network. Two important generalizations of broadcasting are multi-broadcasting and gossiping that require several broadcast operations to be performed in parallel. In designing broadcast protocols, one must take care of both their correctness and their efficiency. Generally speaking, a communication protocol is correct if it allows the messages to be received by the set of all the interested nodes in the network. Besides of the communication scheme (whether broadcasting or multi-casting or what else), the definition of the set of interested nodes depends on a number of assumptions about the network. Among them, the (total or partial or absent) knowledge of each node about the network topology and the (allowed or forbidden) possibility that nodes or links may fail during the execution of the protocol. The main technical challenge in the design of a correct communication protocol is that of avoiding collisions: a collision occurs whenever a node simultaneously receives messages from two or more neighbors. A collision causes the loss of all colliding messages. The efficiency of a communication protocol is usually measured in terms of its completion time, that is, the number of time steps (or rounds) necessary in order to make the messages received by all of the interested nodes. The completion time is usually ex- pressed as a function of the following parameters: number of nodes n, maximum indegree d, diameter D.

25 As far as the completion time of broadcasting protocols is concerned, several upper and lower bounds have been discovered dramatically depending on the network topology and the knowledge of the network owned by the nodes (total knowledge, partial knowledge or no knowledge). Motivated by the applications to dynamic and totally distributed models, recent research has concentrated on unknown networks. In an unknown network, at the beginning of the protocol, each node knows nothing but their own identities (i.e, their own labels). Another interesting feature of the unknown WN is their capability to model a fundamental topic in wireless communication: the Fault-Tolerance. Actually, an unknown WN can be regarded as a known WN in which permanent faults may occur that can be neither foreseen nor recognized. A fault is said to be permanent when it lasts as long as the protocol execution. It follows that a broadcasting protocol designed for an unknown WN is Fault-Tolerant with respect to (both node and link) permanent faults. While the problem of designing eﬃcient fault tolerant broadcasting protocols has already been widely studied for the case of permanent faults, this is not the case both for the more general multi-broadcast operations and for the (more realistic) scenario of dynamic faults occurrence. A fault is said to be dynamic if it may happen and if it may eventually be repaired even during the protocol execution. Our main contribution here has been the design of optimal broadcasting protocols that are fault tolerant even with respect to dynamic faults. Furthermore, a related problem, the Do-All problem, has been considered for a complete network in both cases of permanent and dynamic faults and optimal bounds have been proved for them.

2.2 Energy consumption and connectivity of WNs

More and more attention has been deserved in recent years to ecologic consequences of electromagnetic pollution. Together with economic considerations, such attention is the main motivation to the study and the design of WNs requiring low energy while ensuring a suitable level of connectivity in order to perform some given patterns of communication. In fact, such a problem has been widely considered in the last years with respect to both the pattern of communication that has to be performed in the network and the eﬃciency in performing it. In particular, two patterns of communications have been considered, the gossip (or all-to-all) and the broadcast (or one-to-all) ones. Concerning eﬃciency considerations, it has been taken into account the possibility that a message is required to travel through a bounded number of intermediate nodes before reaching its destination. The importance of this last requirement stands in the quality of service that depends on the number of intermediate nodes (usually 10 nodes for a Radio Network of some hundreds of nodes is a good bound).

More formally, let S = {s1, ..., sn} be a set of points (the radio stations) spread in the plane; we want to assign to each station some power so that the required pattern of communication can be performed and the total power assigned to all stations is minimized. Notice that, if a station si has to send a message to some other station sj, it is not required that si directly transmits to sj, but it may eventually use some intermediate stations as bridges. In other words, we allow multi-hop communications. In order to directly transmit to a station sj, a station si must be assigned with a power P (si) satisfying the following inequality P (si) β ≥ γ, d(si, sj)

26 where d(si, sj) is the distance between si and sj, β ≥ 1 is a parameter depending on environmental conditions, and γ ≥ 1 is a transmission quality parameter. In ideal conditions β = 2, but it may vary from 1 to 6. Once power has been assigned to stations, a communication graph is established: the set of nodes is the set of stations and there is a β (direct) arc from station si to station sj if and only if P (si) ≥ γd(si, sj) .

If the required pattern of communication is the gossip one, we need that each station may send a message to any other station. This translates in requiring that the communication graph is strongly connected. If we further require that each message traverses at most h − 1 intermediate nodes, then we ask for a strongly connected graph of diameter h. The problem of designing a minimum energy strongly connected graph for a set of stations located in the plane (no matter of the diameter) is NP-complete. If the points are located in the 3D space, the problem is even APX-hard. However, a 2-approximation algorithm for the 3D case (and, consequently, for the planar case) and an optimal polynomial-time algorithm for the 1D case (i.e., the stations are located along a line) have been proposed in the literature. As far as a bounded diameter is required, only few results are known. Some tight bounds as functions of the diameter have been proved on the energy consumption in the very special case of the stations uniformly located on a line.

Our contribution in this respect consisted in proving the NP-Completeness of the problem for the symmetric 2D case and the non approximability for the 3D case (simple strong connectivity). As for the case of bounded diameter, we have proposed a 2-approximation algorithm for 1D networks and an approximation algorithm for well-spread 2D networks.

If the required pattern of communication is the broadcast one, we need that one given station, say s, is able to send a message to any other station. This translates in requiring that the communication graph contains a directed tree rooted at s. The problem of designing a minimum energy directed tree rooted at a given node for a set of stations located in the plane (no matter of the diameter) is NP-complete. If the nodes are not points in the plane (that is, if the distance function between pairs of nodes is not a metric), the problem is even non approximable. Instead, an approximation algorithm has been designed for the metric case.

Our contribution for the minimum energy broadcast problem is both theoretical and applicative. From the application point of view, an implementation and experimental test of the approximation algorithm for the metric case has been performed. Aim of the experiments has been the evaluation of the real approximation ratio. They have been performed on a wide number of randomly generated 2-dimensional instances with diﬀerent values for the dimension and the density. From the theoretical point of view we have moved in two directions. First, we have considered simpler instances and tried to study the bounded diameter minimum energy broadcast problem. In this respect, we have designed an optimal polynomial-time algorithm for the case in which stations are located on a line. Secondly, we have considered the general problem, that is, the stations are not located in a metric space, for which we presented a logarithmic approximation algorithm and also tried to face the problem from a diﬀerent perspective. Actually, we have studied the problem in the context of Parameterized Complexity and we have shown its intractability in this last context too.

In the following sections we brieﬂy present the main results obtained.

27 2.3 Optimal F-Reliable Protocols for the Do-All problem on Single-Hop Wireless Networks

In [13] we studied the do-all problem. In this problem, a set of t tasks must be performed by using a synchronous network of p processors. Processors may fail by permanent crashing. We investigate the time and the work complexity of F -reliable protocols for the do-all problem on 1-hop wireless networks without collision detection. An F -reliable protocol is a protocol that guarantees the execution of all tasks if at most F < p faults happen during its execution. Previous results for this model are known only for the case F = p − 1. We obtain the following tight bounds.

• The completion time of F -reliable protocols on 1-hop wireless networks without collision detection is

Ã ( )! t tF √ Θ + min ,F + t . p − F p

• The work complexity of F -reliable protocols on 1-hop wireless networks without collision detection is

Θ(t + F · min{t, F }).

The two lower bounds hold even when the faults only happen at the very beginning of the protocol execution.

2.4 Distributed Broadcast in Radio Networks for Unknown Topology

A multi-hop synchronous radio network is said to be unknown if the nodes have no knowledge of the topology. A basic task in radio network is that of broadcasting a message (created by a ﬁxed source node) to all nodes of the network. Typical operations in real- life radio networks is the multi-broadcast that consists in performing a set of r independent broadcasts. The study of broadcast operations on unknown radio network is started by the seminal paper of Bar-Yehuda et al and has been the subject of several recent works. In [15], we study the completion and the termination time of distributed protocols for both the (single) broadcast and the multi-broadcast operations on unknown networks as functions of the number of nodes n, the maximum eccentricity D, the maximum indegree ∆, and the congestion c of the networks. We establish new connections between these operations and some combinatorial concepts, such as selective families, strongly- selective families (also known as superimposed codes), and pairwise r-diﬀerent families. Such connections, combined with a set of new lower and upper bounds on the size of the above families, allow us to derive new lower bounds and new distributed protocols for the broadcast and multi-broadcast operations. In particular, our upper bounds are almost tight and strongly improve over the previous bounds for a large class of networks.

28 2.5 The Minimum Range Assignment Problem on Linear Radio Networks

Given a set S of radio stations located on a line and an integer h (1 ≤ h ≤ |S| − 1), the min-bounded hops range assignment problem consists in finding a range assignment of minimum power consumption provided that any pair of stations can communicate in at most h hops. Previous positive results for this problem are only known when h = |S| − 1 or when the stations are equally spaced. As for the first case, Kirousis, Kranakis, Krizanc and Pelc (1997) provided an efficient optimal algorithm while, for the second case, they derive an efficient approximation algorithm. In [17] we present the first polynomial time, approximation algorithm for the min- bounded hops range assignment problem. The algorithm guarantees an approximation ratio of 2 and runs in O(hn3) time. We also prove that, for fixed h and for “well spaced” instances (a broad generaliza- tion of the uniform chain case), the problem admits a polynomial-time approximation scheme (PTAS). This result significantly improves over the approximability result given by Kirousis et al. Both our approximation results are obtained by new algorithms that exactly solves two natural variants of the min-bounded hops range assignment problem that may have independent interest: the problem in which every station must reach a fixed one in at most h hops and the problem in which the goal is to select a subset of Bases among the stations such that all the other stations must reach one of Bases in at most h − 1 hops. Finally, we show that for h = 2 the min-bounded hops range assignment problem can be exactly solved in O(n3) time.

2.6 The Minimum Broadcast Range Assignment Problem on Linear Multi-Hop Wireless Networks

Given a set S of radio stations located on a line, a station s ∈ S and an integer h (1 ≤ h ≤ |S| − 1), the min-bounded hops broadcast range assignment problem consists in finding a range assignment of minimum power consumption provided that s can broadcast a message to any other station in at most h hops. In [14] we present a polynomial-time algorithm for the min-bounded hops broadcast range assignment problem running in O(hn3) time. More precisely, we first prove a structural property of optimal solutions, and then we exploit it in order to derive a dynamic programming algorithm able to find minimum energy h-hops broadcast range assignment. Independently, a polynomial time algorithm for the unbounded case (h = |S| − 1) is presented in [11] using similar techniques with those in [14].

2.7 Energy Eﬃcient Broadcasting in General Wire- less Networks

The energy-eﬃcient broadcast problem can be thought of as the following graph-theoretic problem: Given a complete directed graph G with n nodes and with costs associated with its edges, and a special node s, ﬁnd a minimum weight assignment to the nodes of G so that the corresponding transmission graph (the node-induced subgraph which has an

29 edge (u, v) of G only if the weight of node u is larger than the cost of the edge (u, v)) is a tree directed out of s and spanning all the nodes of the graph. The problem has been proved to be³ NP-hard´ and, furthermore, inapproximable within (1 − ²) ln n unless NP ⊆ DTIME no(ln ln n) , through approximation preserving reductions from Set Cover and Connected Dominating Set. In [11, 12], we present a logarithmic approximation algorithm for the problem, thus, we achieve the best possible approximation ratio within constant factors. As an intermediate step of the algorithm, we devise a novel reduction to a variant of a the Connected Dominating Set Problem, namely, the Node-Weighted Connected Dominating Set Prob- lem. The energy eﬃcient broadcast problem can be seen as a member of a family of interesting wireless network design problems which can be obtained if one considers diﬀerent connectivity requirements for the transmission graph. We plan to work on such problems in future work.

2.8 Some Recent Theoretical Advances and Open Questions on Energy Consumption in Ad-Hoc Wireless Networks

One of the main benefits of power controlled ad-hoc wireless networks is their ability to vary the range in order to reduce the power consumption. Minimizing energy consumption is crucial on such kind of networks since, typically, wireless devices are portable and benefit only of limited power resources. On the other hand, the network must have a sufficient degree of connectivity in order to guarantee fast and efficient communication. These two aspects yield a class of fundamental optimization problems, denoted as range assignment problems, that has been the subject of several works in the area of wireless network theory. Thus, the primary aim in [16] was to describe the most important recent advances on this class of problems. Rather than completeness,in [16] we try to provide results and techniques that seem to be the most promising to address the several important related problems which are still open. Discussing such related open problems are indeed our other main goal.

2.9 On the Approximation Ratio of the MST- based Heuristic for the Energy-Eﬃcient Broad- cast Problem in Static Ad-Hoc Radio Networks

In [9] we present a new analysis of the approximation ratio of the MST-based heuristic for the Minimum Energy Broadcast Problem in Ad-Hoc Radio Networks. This fundamental problem is known to be NP-hard and approximable within a worst-case constant ratio by simply computing the Minimum Spanning Tree (MST) of the graph underlying the wireless network (i.e., by the MST-based heuristic introduced by Ephremides et al.). However, the best known theoretical upper bound on this ratio is very large: 12. Our intuition here is that this large value is due to a too rough and pessimistic scenario considered by the previous worst-case analysis of the MST-based heuristic. We use

30 techniques from to derive a polynomial-time computable lower bound on the optimal cost of any instance of the problem. Thanks to this lower bound, we were able to evaluate the approximation ratio over thousands of random instances (i.e. instances in which nodes are chosen uniformly and independently at random), for several values of the network size n and the density. The previous experimental studies on this problem were only able to compare a set of heuristics, one to each other, on random instances. The main result of this paper is that, in all the experimental tests, the approximation ratio has never achieved a value greater than 6.4. Furthermore, the worst (i.e. the largest) values of this ratio are achieved for small network sizes (i.e. n ≤ 9). This is consistent with some previous works on the asymptotical properties of Euclidean MST’s. We also provide a clear geometrical motivation of such good approximation results, i.e., the main reasons for which the ratio 12 is not tight (at least) for the adopted input model.

2.10 Minimum-Energy Parameterized Problems in (Wireless Network) Graphs

Parameterized Complexity has been introduced by Downey and Fellows about ten years ago. It is a powerful framework with which to address the different “parameterized behavior” of many computational problems. Almost all natural problems have instances consisting of at least two logical items; many NP-complete problems admit “efficient” algorithms for small values of one item (the parameter). A parameterized problem is said to be fixed parameter tractable if it admits a solving algorithm whose running time on instance (x, k) is bounded by f(k) · |x|α, where f is an arbitrary function and α is a constant not depending on the parameter k. The class of fixed parameter tractable problems is denoted by FPT. In order to characterize those problems that do not seem to admit a fixed parameter efficient algorithm, Downey and Fellows defined a fixed parameter reduction and a hierarchy of classes W [1] ⊆ W [2] ⊆ ... including likely fixed parameter intractable problems. Each W-class is the closure under fixed parameter reductions with respect to a kernel problem, which is usually formulated in terms of special mixed-type boolean circuits in which the number of input lines set to true is bounded by a function of the parameter. In [18] we prove the parameterized hardness of minimum energy broadcast in general graphs. In particular, we prove that, given a weighted graph G with edge weight w, a source node s ∈ V (G) and a parameter k, it is W [2]-complete to decide whether G P contains a spanning tree T rooted at s such that u∈V (G) max(u,v)∈E(T ){w(u, v)} ≤ k. The study of such a problem is motivated by a result by Clementi et al. showing that the Minimum Energy Broadcast Problem is not approximable in general graphs.

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