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Distributed Algorithms for Edge Dominating Sets Distributed Algorithms for Edge Dominating Sets Jukka Suomela Helsinki Institute for Information Technology HIIT, University of Helsinki P.O. Box 68, FI-00014 University of Helsinki, Finland [email protected].fi ABSTRACT By definition, any maximal matching is an edge dominat- An edge dominating set for a graph G is a set D of edges ing set. An edge dominating set is not necessarily a match- such that each edge of G is in D or adjacent to at least one ing; however, given an edge dominating set D, it is straight- edge in D. This work studies deterministic distributed ap- forward to construct a maximal matching with at most jDj proximation algorithms for finding minimum-size edge dom- edges [25]. Hence a minimum maximal matching (a maxi- inating sets. The focus is on anonymous port-numbered net- mal matching with the smallest possible number of edges) is works: there are no unique identifiers, but a node of degree also a minimum edge dominating set. d can refer to its neighbours by integers 1; 2; : : : ; d. The This is a corollary of a more general result due to Allan present work shows that in the port-numbering model, edge and Laskar [1]: if a graph is claw-free (no induced subgraph dominating sets can be approximated as follows: in d-regular K1;3), then a minimum maximal independent set is also a graphs, to within 4 − 6=(d + 1) for an odd d and to within minimum dominating set. The line graph L(G) of any graph 4−2=d for an even d; and in graphs with maximum degree ∆, G is claw-free, the dominating sets of L(G) correspond to the to within 4 − 2=(∆ − 1) for an odd ∆ and to within 4 − 2=∆ edge dominating sets of G, and the maximal independent sets for an even ∆. These approximation ratios are tight for all of L(G) correspond to the maximal matchings of G. values of d and ∆: there are matching lower bounds. 1.2 Centralised Polynomial-Time Algorithms From the perspective of centralised polynomial-time algo- Categories and Subject Descriptors rithms, the problem of finding a minimum edge dominating C.2.4 [Computer-Communication Networks]: Distrib- set is equivalent to the problem of finding a minimum maxi- uted Systems; F.2.2 [Analysis of Algorithms and Prob- mal matching. Both of these are NP-hard optimisation prob- lem Complexity]: Nonnumerical Algorithms and Prob- lems [25], and they are hard to approximate to within factor lems|computations on discrete structures 7=6 − [9]. The problem of finding a minimum-weight edge cover is as hard to approximate as minimum-weight vertex cover [8]. General Terms The connection between matchings and edge dominating Algorithms, Theory sets implies a simple 2-approximation algorithm: any max- imal matching is a 2-approximation of a minimum edge dominating set. Approximating minimum-weight edge dom- 1. INTRODUCTION inating sets is less straightforward, but Fujito and Naga- This work studies the approximability of the edge dom- mochi [12] show how to find a 2-approximation. Polynomial- inating set problem from the perspective of deterministic time approximation schemes are known for planar graphs [6] (non-randomised) distributed algorithms in anonymous port- and civilised graphs [15]. numbered networks. 1.3 Distributed Algorithms 1.1 Edge Dominating Sets and Matchings Edge dominating sets have received little attention in the Let G be a simple, undirected graph with the edge set EG . distributed computing community. However, some results A set D ⊆ EG of edges is an edge dominating set for G if related to matchings and independent sets have straightfor- each edge e 2 EG n D is adjacent to at least one edge in D. See Figure 1 for examples. (a) (b) (c) (d) c ACM, 2010. This is the author’s version of the work. It is posted here by permission of ACM for your personal use. Not for redis- Figure 1: (a) An edge dominating set. (b) A max- tribution. The definitive version was published in Proc. 29th ACM imal matching and hence another edge dominating SIGACT-SIGOPS Symposium on Principles of Distributed Computing set. (c) A minimum edge dominating set. (d) A min- (PODC’10, July 25–28, 2010, Zurich, Switzerland). imum maximal matching and hence another mini- http://doi.acm.org/10.1145/1835698.1835783 mum edge dominating set. ward corollaries that concern the distributed approximabil- Lower bound Graph family Approx. ratio Time ity of edge dominating sets. Upper bound On the positive side, one can again take any algorithm d-regular graphs: that finds a maximal matching and apply it to find a 2- 6 approximation of a minimum edge dominating set. For ex- d = 1; 3;::: 4 − Theorem 2 O(d2) ample, if we have unique node identifiers in the network, d + 1 Theorem 4 we can use the deterministic algorithms by Ha´n´ckowiak et 2 d = 2; 4;::: 4 − Theorem 1 O(1) al. [14] and Panconesi and Rizzi [19], with running times d Theorem 3 O(log4 n) and O(∆ + log∗ n) communication rounds, respec- tively; here n is the number of nodes and ∆ is the maximum graphs with maximum degree ∆: degree. ∆ = 1 1 trivial O(1) The running times of these algorithms depend on n, and trivial this is unavoidable if we want to achieve an approximation 2 factor better than 3. Czygrinow et al. [10] and Lenzen and ∆ = 3; 5;::: 4 − Corollary 1 O(∆2) Wattenhofer [17] show that finding a constant-factor approx- ∆ − 1 Theorem 5 imation of a maximum independent set in a cycle requires 2 Corollary 1 2 ∗ ∆ = 2; 4;::: 4 − O(∆ ) Ω(log n) communication rounds, and a simple local reduc- ∆ Theorem 5 tion [22] gives the same lower bound for finding a factor 3− approximation of a minimum edge dominating set. Table 1: Approximability of edge dominating sets: the best possible approximation ratios that can be 1.4 Algorithms in Port-Numbered Networks achieved by any deterministic distributed algorithm The above results deal with deterministic distributed algo- in the port-numbering model. rithms in networks with unique node identifiers. This work studies a strictly weaker model of computation: determin- istic distributed algorithms in anonymous networks in the nodes may look identical from the perspective of a distrib- port-numbering model: there are no node identifiers, but uted algorithm, but all edges do not look identical to each a node of degree d can refer to its neighbours by integers other. Tight lower bound constructions for covering prob- 1; 2; : : : ; d. See Section 2 for a formal definition of the model. lems such as vertex covers and dominating sets are typically Computation in synchronous port-numbered networks has trivial: a cycle or a complete graph will do. This is not the been studied for decades; one of the pioneers was Angluin [2] case with edge-based problems. in 1980. However, the main focus has been on global prob- Second, this work gives yet another example of the close lems such as leader election [2, 24], construction of span- connection between the port-numbering model and local al- ning trees [24], computation of functions that depend on all gorithms. In a strictly local algorithm, the running time nodes [5, 7, 23], recognition of topological properties [2, 24], does not depend on the number of nodes [18, 22]. Even and graph exploration and rendezvous [16]. Such problems though the negative results hold for any algorithm, regard- typically require Ω(n) communication rounds { or, in many less of its running time, the matching positive results are cases, are unsolvable in the port-numbering model. local algorithms: the running times depend on the parame- Much less is known about graph problems that are of a ters d and ∆, but they are independent of n. Indeed, these more local nature and have potential for efficient, highly algorithms are the best known deterministic local algorithms scalable distributed algorithms. Classical packing problems for the edge dominating set problem { it is not known if a such as matchings and independent sets are typically un- better approximation ratio can be achieved in constant time solvable for trivial reasons, but covering problems are more with the help of unique node identifiers. promising. Node-based covering problems (the task is to choose a subset of nodes that \covers" the graph) have been studied in prior work: for example, the vertex cover problem 2. PRELIMINARIES can be approximated within factor 2 in the port-numbering Let G be a simple, undirected graph with the node set VG model in bounded-degree graphs [3, 4], and this approxima- and the edge set EG . An edge e = fu; vg 2 EG is said to tion guarantee is tight. However, it seems that edge-based cover the nodes u and v, and an edge e1 2 EG is said to covering problems have not been studied previously in this dominate any edge e2 2 EG that is adjacent to e1, including model. e1 itself. These terms are generalised to sets of edges and nodes in a natural manner: for example, a set C ⊆ EG of 1.5 Contributions edges covers a set of nodes X ⊆ VG if for each v 2 X there The contributions are summarised in Table 1. This work is an e 2 C that covers v. An edge cover is a set C ⊆ EG presents a complete characterisation of the deterministic ap- that covers VG and an edge dominating set is a set D ⊆ EG proximability of edge dominating sets in the port-numbering that dominates EG . model, both in graphs of maximum degree ∆ and in d- A set M ⊆ EG is a matching if each node v 2 VG is regular graphs, for all values of the parameters ∆ and d.
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