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 ap- poximations . 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 complex- ity 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, inter- esting 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 jΦ(u) ¡ Φ(v)j ¸ 2 if D(u; v) = 1 and jΦ(u) ¡ Φ(v)j ¸ 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 º = maxv2V Φ(v) ¡ minu2V 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 : fu; vg 2 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 Â(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) = Â(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 interest- ing 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 ¸ colors for a planar graph G, when ¸ ¸ 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 ¸-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 ¸ 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(∆ ).