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Approximating the Independent Domatic Partition Problem in Random Geometric Graphs – an Experimental Study

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Approximating the Independent Domatic Partition Problem in Random Geometric Graphs – an Experimental Study CCCG 2010, Winnipeg MB, August 9–11, 2010 Approximating the Independent Domatic Partition Problem in Random Geometric Graphs – An Experimental Study Dhia Mahjoub ∗ Angelika Leskovskaya † David W. Matula ∗ Abstract often modeled in practice as Random Geometric graphs (RGGs). A random geometric graph G(n, r) is defined We investigate experimentally the Domatic Partition by n vertices uniform in the unit square with an edge be- (DP) problem, the Independent Domatic Partition tween any two vertices of V within Euclidean distance (IDP) problem and the Idomatic partition problem in r of each other. An RGG simply induces a uniform Random Geometric Graphs (RGGs). In particular, probability distribution on a Unit Disk Graph (UDG). we model these problems as Integer Linear Programs A variation of the DP problem is the Independent Do- (ILPs), solve them optimally, and show on a large set matic Partition (IDP) problem which seeks to partition of samples that RGGs are independent domatically full a graph G into disjoint independent dominating sets. most likely (over 93% of the cases) and domatically full The independent domatic number dind(G) is the maxi- almost certainly (100% of the cases). We empirically mum size of such a partition. confirm using two methods that RGGs are not idomatic For any graph G, dind(G) d(G) δ(G) + 1 where on any of the samples. We compare the results of the δ(G) denotes the minimum≤ degree in≤ G. If d(G) = ILP-based exact algorithms with those of known color- δ(G) + 1 and/or dind(G) = δ(G) + 1, then G is called ing algorithms both centralized and distributed. Color- domatically full and/or independent domatically full re- ing algorithms achieve a competitive performance ratio spectively [4]. A graph whose vertices V can be strictly in solving the IDP problem [12, 11]. Our results on the partitioned into disjoint independent dominating sets is high likelihood of the “independent domatic fullness” termed indominable [2] or idomatic [4]. The idomatic of RGGs lead us to believe that coloring algorithms can number id(G) is the partition’s maximum size. be specifically enhanced to achieve a better performance The study described herein is motivated by the desire ratio on the IDP size than [12, 11]. We also investigate to empirically verify the existence of the upper bound experimentally the extremal sizes of individual domi- of δ + 1 disjoint independent dominating sets in RGGs nating and independent sets of the partitions. (which model Wireless Sensor Networks). Namely, are random geometric graphs independent domatically full 1 Introduction and Motivation in practice? A k coloring of a graph G = (V, E) is a partition − The domatic partition (DP) problem is a classical prob- Π = V1,V2, ..., Vk of the vertex set V (G) into indepen- lem in graph theory whose goal is to partition a graph dent sets Vi, each of which is called a color class. A G into disjoint dominating sets. The domatic num- vertex v Vi is called colorful if each color 1 i k ber d(G) is the maximum number of dominating sets appears on∈ the closed neighborhood of v. A k-coloring≤ ≤ f in such a partition [4]. The concept has various appli- is called a fall k-coloring if every vertex in f is colorful. cations such as the strategic placement of objects on Clearly, a strict partition of the vertex set V (G) into the nodes of a network (facility location) [18, 3] or the k independent dominating sets (idomatic partition) is identification of the maximum number of disjoint trans- equivalent to finding a fall k-coloring of G [9]. We note mitting groups in a network [7]. Recently, the DP prob- that if V1,V2, ..., Vδ+1 are disjoint independent dominat- lem found applications for efficient replica placement δ+1 ing sets of G, then the induced subgraph Si=1 Vi is a in Peer-to-Peer systems and for cooperative caching be- maximal idomatic subgraph of G of the same minimum tween ISPs to improve video on-demand delivery strate- degree. gies [19, 1]. More relevant to our work is the application Moreover, we experimentally study the “domination to energy conservation and sleep scheduling in Wireless chain” γ(G) i(G) β0(G) in RGGs. The “domina- Sensor Networks (WSN) [17, 16, 8, 10, 12, 11] which are tion chain” is≤ a relation≤ between graph parameters that is satisfied in any graph G [4], where γ(G) is the size of ∗Department of Computer Science, Southern Methodist University. Dallas, TX 75275-0122, the minimum dominating set (MDS) termed the domi- {dmahjoub,matula}@lyle.smu.edu nation number, i(G) is the size of the minimum indepen- †Engineering Management, Information, and Systems Depart- dent dominating set (MIDS) termed the independence ment, Southern Methodist University. Dallas, TX 75275-0122, domination number and β (G) is the size of the maxi- [email protected] 0 22nd Canadian Conference on Computational Geometry, 2010 k k mum independent set (MaxIS) termed the independence xv + xu 1 u,v V : (u,v) E, k K (2) number. Finding these values are NP-complete prob- δ+1 ≤ ∀ ∈ ∈ ∈ k X xu 1 u V (3) lems in general graphs and Unit Disk Graphs [4, 13]. ≤ ∀ ∈ The decision version of the DP problem for k 3 k=1 k ≥ uk 0, 1 , x 0, 1 u V, k K (4) is NP-complete in general graphs, circular arc graphs, ∈ { } u ∈ { } ∀ ∈ ∈ split graphs and bipartite graphs. It is in P for interval k where uk=1 if dominating set Sk = u x = 1 graphs, proper circular arc graphs and strongly chordal { | u } graphs [3, 7]. It is an open question whether the problem is selected in the IDP and uk=0 otherwise. Constraint is also NP-complete in Unit Disk Graphs, but it most (1) expresses domination, (2) independence, (3) node likely is [14]. The IDP and Idomatic partition (fall col- disjointness, i.e. a node can be part of at most one oring) problems are NP-complete in general graphs and set, and (4) variable integrality. We also formulate k-regular and bipartite graphs as well [5, 9]. We know the idomatic partition problem as an ILP where we of two constant factor approximation algorithms to the maximize the size of the independent domatic partition domatic partition [17] and connected domatic partition as well as the total number of packed nodes in the [16] problems in Unit Disk Graphs (UDG). In [12, 11] sets of the partition. The exact algorithms for the we show competitive performance ratios by using graph MDS, MIDS and MaxIS problems are also modeled coloring algorithms to empirically approximate the DP as ILPs. For illustrative purposes, we present the ILP and IDP problems. formulation of MIDS below: minimize x 2 Our Contributions X u u V ∈ s.t. xu+ X xv 1 u V (1) In this paper, our main contributions are: ≥ ∀ ∈ v:(u,v) E -We solve the IDP problem optimally and show that ∈ xu + xv 1 u,v V : (u,v) E (2) over 93% of the RGG instances are independent domat- ≤ ∀ ∈ ∈ ically full and 100% of the instances are domatically xu 0, 1 u V (3) full. The high likelihood of the existence of an opti- ∈ { } ∀ ∈ mal partition of δ + 1 independent dominating sets in typical RGGs suggests that coloring algorithms can be fine-tuned to achieve a better performance ratio [12, 11]. Coloring Heuristics. In this study, in order to -We confirm by Smallest Last (SL) coloring [15] for experimentally approximate the IDP problem, we use a large sample of RGG instances that χ(G) ω(G) > 5 centralized graph coloring heuristics: Smallest Last, δ(G) + 1, hence these graphs cannot be idomatic≥ [2]. Largest First, Lexicographic, Radial Sweep and Ran- In addition, we formulate the idomatic partition prob- dom. These algorithms are described in detail in [12]. lem as an ILP and confirm through experiments that all We also experiment with 4 distributed coloring heuris- graphs of the sample are not idomatic. tics: Trivial Greedy, Largest First, Lexicographic, and -We experimentally study the node packing in the sets 3Cliques-Last. We discuss these algorithms with ample of the IDP solution and also report on the domination detail in [11] . chain values and compare the results obtained by ILP algorithms and coloring algorithms with the asymptotic 4 Experimental Results bounds based on “optimal” triangular lattice packing. We believe this study answers relevant questions for In this paper, ILP models are solved optimally using practitioners and also stimulates further research on the CPLEX 10.0 installed on a Dual Quad Core Intel Xeon approximability of the IDP problem in UDGs and RGGs X5570 with 72 GB RAM running CentOS Linux 2.6.18. and on the asymptotic behavior of domination and do- Each core is clocked at 3.00 GHz. The coloring algo- matic properties in RGGs. rithms are implemented in C#.Net (Microsoft Visual Studio 2005) on an Intel Core 2 Duo E8400 processor clocked at 3.00 GHz with 3 GB RAM running Win- 3 Algorithms dows Vista Enterprise SP1. Our data set consists of IDP Formulation. Given a graph G = (V, E) and the 15 graphs generated randomly with δ 5,10,20 and n 50,100,200,400,800 . Results for each∈{ (δ, n)} pair set K = 1, ..., δ + 1 , we formulate the IDP problem as ∈{ } the following{ Integer} Linear Program (ILP): are averaged over 20 RGG instances, except for the δ+1 (δ = 20, n = 800) case where we average over 10 in- maximize X uk stances, given that the running times of the ILP models k=1 were prohibitively long.
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