839 Distributed Weighted Vertex Coer ia Maximal Matchings Fabrizio Grandoni 1, Jochen Könemann 2, and Alessandro Panconesi 1 1 Dipartimento di Informatica, Uniersità di Roma La Sapienza Via Salaria 113, 00198 Roma, Italy {grandoni,ale}@di.uniroma1.it 2 Department of Combinatorics and Optimization, Uniersity of Waterloo 200 Uniersity Aenue West, Waterloo, ONN2L 3G1, Canada jochen@math.uwaterloo.ca Abstract. In this paper we consider the problem of computing a minimumweight ertex-coer in an n-node, weighted, undirected graph G =V, E). We present a fully distributed algorithm for computing ertex coers of weight at most twice the optimum, in the case of integer weights. Our algorithm runs in an expected number of Ologn + logŵ ) communication rounds, where Ŵ is the aerage ertex-weight. The preious best algorithm for this problem requires Olognlogn + logŵ )) rounds and it is not fully distributed. For a maximal matching M in G it is a well-known fact that any ertex-coer in G needs to hae at least M ertices. Our algorithm is based on a generalization of this combinatorial lower-bound to the weighted setting. 1 Introduction We are gien an undirected graph G =V,E) and non-negatie ertex weights w W for all ertices V. Aertex coer is a subset C V such that each edge e E has at least one end-point in C. Intheminimum-weight ertex-coer problem we want to compute a ertex-coer of smallest total weight. Computing minimum-weight ertex-coers is NP-hard [4]. Papadimitriou and Yannakakis [14] show that the problem is APX-hard. More recently, Håstad [8] proes that there is no 7/6 ε)-approximation algorithm for the ertex-coer problem for any ε > 0 unless P = NP. On the positie side, the best known algorithms for the ertex coer problem are due to Bar-Yehuda and Een [1], and to Monien and Speckenmeyer [12]. These algorithms achiee an approximation ratio of 2 loglogn 2logn ). In graphs with maximum degree, Hochbaum [9] gies a 2 1/ )-approximation algorithm for the problem. This was subsequently improed by Halldórsson and Radhakrishnan [5] who present a 2 log )+O1) )-approximation algorithm. Finally, Halperin [6] presents the currently best algorithm for the problem with a performance ratio of 2 1 o1)) 2lnln ln ). In the distributed setting, it is known how to compute a 2-approximate ertex coer in the unweighted case. This can be achieed by computing a maximal matching in The first and third authors were supported with funding from EC Projects DELIS and EYES, and project WebMinds of the Italian Ministry of Uniersity and Research MIUR) L. Wang Ed.): COCOON 2005, LNCS 3595, pp. 839 848, 2005. c Springer-Verlag Berlin Heidelberg 2005
840 Fabrizio Grandoni, Jochen Könemann, and Alessandro Panconesi the graph and by including the matched nodes in the coer. A maximal matching can be computed in Olog 4 n) rounds ia the algorithm of Hanckowiack et al. [7], and in O + log n) rounds ia the algorithm of Panconesi and Rizzi [13]. Both algorithms are deterministic. Maximal matchings can also be computed in an expected number of Ologn) rounds ia the randomized algorithm of Israeli and Itai [10]. For the weighted case a 2 + ε)-approximation can be computed deterministically in Olognlog 1 ε ) many rounds by using the algorithm of Khuller et al. [11]. Their algorithm is stated as a PRAM algorithm, but it is readily seen to be a bona fide distributed algorithm. Let Ŵ be the aerage weight. Then, by setting ε = 1/nŴ + 1), the latter algorithm computes a 2-approximate ertex coer in Olognlogn + logŵ )) communication rounds. Note that the aboe choice of ε requires global knowledge of the quantity nŵ. This assumption may not be realistic in all scenarios. In this paper we present an improed fully-distributed algorithm to compute a 2- approximate weighted ertex coer, in the case of integer weights. Our main result can be stated as follows: Theorem 1. There is a fully distributed algorithm which computes a 2-approximate weighted ertex coer in an expected number of Ologn + logŵ) communication rounds. The message size is OlogW) and the local computation done in each round is O log W)) in expectation. Our algorithm can be iewed as a generalization of the reduction from unweighted ertex coer to maximal matching. The basic idea is to expand each node of weight w into w micro-nodes 1),2)...,w ), and connect each i) to eery u j) wheneer u is an edge of the network. Then a maximal matching in the auxiliary graph is computed. The ertex coer is gien by the nodes such that all the corresponding micronodes are matched. If the maximal matching is computed ia the fully-distributed algorithm of Israeli and Itai, the algorithm halts in an expected number of Ologn + logŵ) rounds. A naie implementation of the matching algorithm by Israeli and Itai leads to pseudo-polynomial message and time complexity in each round. The main insight leading to the bounds on message-size and local computation time in Theorem 1 is to keep an implicit representation of the auxiliary graph and a maximal matching in it. The rest of this paper is organized as follows. In Section 2 we introduce some preliminaries. Our algorithm relies on a careful adaptation of the matching algorithm by Israeli and Itai. We present this adaptation in Section 3. Finally, Sections 4 and 5 deal with the naie and refined ersions of our weighted ertex coer algorithm, respectiely. 2 Preliminaries The minimum-weight ertex coer problem can be formulated as an integer linear program ILP): min V w x s.t. x + x u 1, u E; x {0,1}, V.
Distributed Weighted Vertex Coer ia Maximal Matchings 841 Each assignment of the ariables which satisfies the constraints feasible solution) corresponds to a ertex coer containing exactly the nodes with x = 1. By LP) we denote the natural linear programming relaxation of ILP). Let N) be the set of neighbors of.thelinear programming dual D)ofLP)is: max u E y u s.t. u N) y u w, V ; y u 0, u E. By weak duality e.g., see [2]), the alue of each feasible solution of D) is a lower bound for the alue of eery feasible solution of LP) and hence IPL). In this paper we consider a synchronous message-passing model of computation. The computation proceeds in rounds. In each round, a node can send/receie a message of unbounded size) to/from each one of its neighbors, and execute an unbounded amount of computation. No global knowledge is aailable including the number n of nodes in the graph). The algorithms presented can be easily modified so as to work in a non-faulty) asynchronous system also. By Bp), p [0,1], we denote a 0-1 Bernoulli random ariable, which takes alue one with probability p. Arandom bit is a Bernoulli ariable B0.5). 3 Distributed Maximal Matching A matching of a graph G =V,E) is a subset M E such that no two edges of M are incident to the same node. The results of the next sections are based on the following simplified ersion M of the distributed maximal-matching algorithm of Israeli and Itai [10]. Algorithm M works in phases, each one consisting of a constant number of rounds. In each phase, a matching is computed and the edges incident on matched nodes are remoed. The algorithm halts when no edge is left. The maximal matching is gien by the union of the matchings found in the different phases. In a gien phase a matching is computed in the following way. Let G =V,E ) be the current graph. By N ) and δ we denote the set of neighbors of and the degree of in G, respectiely. Each node randomly decides to be a sender or a receier with probability one half. Note that the same node may play a different role in different phases. Each sender u selects one neighbor N u) uniformly at random and makes a proposal to. Each receier which receies at least one proposal, selects one of the proponents arbitrarily) and accepts its proposal. The matching is gien by the edges corresponding to accepted proposals. Let a node be good if at least one third of its neighbors u hae degree δ u δ.to proe the bound on the number of rounds, we use the following simple combinatorial result [10]: Lemma 1. At least one half of the edges of a graph are incident to good nodes. Theorem 2. Algorithm M computes a maximal matching in Ologn) expected rounds. Proof. The correctness of the algorithm is triial.
842 Fabrizio Grandoni, Jochen Könemann, and Alessandro Panconesi We show that in each phase at least a constant fraction of the edges is remoed in expectation. This implies that the expected number of rounds is Ologn 2 )) = Ologn). Consider a good node of G in a gien phase. The probability P that accepts a proposal is lower bounded by: P 1 1 2 1 1 ) ) u N 2δ ). u From the definition of good nodes: 1 1 ) u N 2δ ) u u N ):δ u δ 1 1 ) 2δ 1 1 ) δ /3 2δ e 1/6. Thus P 1 e 1/6 )/2 and the expected number of edges remoed is at least a fraction 1 e 1/6 )/4 of the total. 4 Distributed Vertex Coer ia Maximal Matchings In this section we present a simple pseudo-polynomial reduction from the problem of computing a 2-approximate ertex coer to the problem of computing a maximal matching in an auxiliary graph. Thanks to this reduction, a 2-approximate ertex coer can be computed in Ologn + logŵ) expected rounds ia algorithm M of section 3. Consider the following auxiliary graph G. For each node of G, G contains w micro-nodes 1),2)...w ). Two micro-nodes i) and u j) are adjacent if and only if u is an edge of G. In Figure 1 an example of the reduction is gien. 3 2 1 Fig. 1. A weighted graph G on the left) with the corresponding auxiliary graph G. A maximal matching M of G is indicated ia dashed lines. The dashed nodes of G form a ertex coer Let M be a maximal matching in G. ByVM) we denote the set of nodes of G such that all the corresponding micro-nodes i) are matched by M. Lemma 2. Set VM) is a 2-approximate ertex coer of G. Proof. Assume by contradiction that V M) is not a ertex coer. Thus there are two adjacent nodes and u in G which do not belong to V M). This implies that there are two adjacent micro-nodes i) and u j) in G which are not matched by M. Then the set M = M {i)u j)} is a matching, which contradicts the maximality of M.
Distributed Weighted Vertex Coer ia Maximal Matchings 843 Let apx and opt denote the weight of the ertex coer found and that of a minimum weight ertex coer, respectiely. Moreoer, let z be the number of micro-nodes in {1),2)...w )} that are matched by M. A feasible solution of D) is obtained by assigning to each dual ariable y u the number of edges of the kind i)u j) M. This solution is feasible since, for eery V: u N) y u = z w. By weak duality we obtain apx V z 2 u E y u 2opt and hence V M) is 2-approximate. Lemma 2 suggests a strategy to compute a 2-approximate ertex coer distributiely. The idea is to simulate the behaior of algorithm M on a irtual auxiliary graph G, and then to select the nodes in the ertex coer as suggested by Lemma 2. Specifically, each node simulates the execution of the algorithm on the corresponding micro-nodes i) in G. Wheneer two micro-nodes i) and u j) of G need to communicate, nodes and u are responsible for allowing such communication. The ertex coer is gien by the nodes such that all the corresponding micro-nodes i) are matched by the maximal matching computed. Since the irtual auxiliary graph contains OnŴ) nodes, the total number of rounds is OlognŴ)) = Ologn + logŵ). This naie application of Lemma 2 has two major drawbacks. The first problem is the large message size. In fact, in each phase all the micro-nodes of maysenda proposal to some micro-node of u. Thus the message size is ΩW). A second problem is the time complexity of the algorithm: consider a node in a gien phase. Each micro-node i) of, with probability one half, needs to select one neighbor out of Θ W) uniformly at random. This random selection can be performed in Θlog W)) expected time, assuming that the cost of generating a random bit is O1) e.g., see [3]). Thus the expected time complexity of each phase is ΩW log W)). In next section we show how to sole both problems by creating the matchings implicitly. 5 An Improed Algorithm In this section we present an improed fully distributed algorithm A for computing a 2- approximate ertex coer. Algorithm A still requires Ologn+logŴ) expected rounds, but it reduces the size of the messages to OlogW) and the expected time complexity of each phase to O log W )). The basic structure of algorithm A is analogous to the structure of the naie algorithm described in preious section: in each phase, a matching in the current auxiliary graph G is computed, and the matched nodes are remoed from G together with all the edges incident to them). The algorithm halts when no edge is left. The ertex coer is gien by the nodes such that all the corresponding micro-nodes i) are matched by one of the matchings computed. The main noelty in Algorithm A is that matchings are created implicitly: in each phase each node only knows the number of the corresponding matched micro-nodes. Intuitiely, this simplification is allowed by the symmetry properties of G: all the micro-nodes corresponding to a node hae the same degree and share the same neighborhood. This inariant is kept by all the induced subgraphs of G. Algorithm A, which is described in Figure 2, works in phases. Each phase consists of a constant number of communication rounds. Each node has an associated state
844 Fabrizio Grandoni, Jochen Könemann, and Alessandro Panconesi w = w ; N )=N), s = actie; while s = actie) { send w and receie w u to/ from all u N ); N )={u N) : w u > 0}; if N ) = 0) s = outside; else { compute the proposals p u); send p u) and receie p u ) to/ from all u N ); compute the counter-proposals c u); send c u) and receie c u ) to/ from all u N ); for all u N )) w = w c u) c u ); if w = 0) { send w to all u N ); s = inside; } } } Fig. 2. Protocol for node for 2-approximate ertex coer s, which is initially actie. In each phase, part of the actie nodes switch to the state inside or outside, and the algorithm terminates when no actie node is left. When a node leaes the actie state, it halts. At the end of the algorithm, the inside nodes form a ertex coer. In more details, each node has an associated residual weight w, which is initially w. The residual weight w can be interpreted as the number of micro-nodes i) of in the current auxiliary graph G. Note that all the micro-nodes i) haethesamedegree W = u N) w u. In each phase, the expected residual weight of actie nodes decreases. The decrease of w in a gien phase reflects the number of micro-nodes of that hae been matched in that phase. At the beginning of each phase, each actie node sends w to all its currently actie neighbors N ). The neighbors with w u = 0 are remoed from N ).IfN ) becomes empty, node switches to the outside state. In fact, in this case the degree W of the micro-nodes i) is zero, and thus they will neer be matched. Otherwise, sends a proposal p u) to each actie neighbor u N ). Thealue of p u) can be interpreted as the number of proposals directed from the micro-nodes of to the micro-nodes of u.letp be the sum of the proposals p u): p = u N ) p u). This quantity can be iewed as the number of micro-senders among 1),2)...w ). We postpone a detailed description of how proposals are fixed until later. For each proposal p u ) receied, node replies with a counter-proposal c u).the counter-proposal c u) can be interpreted as the number of micro-nodes of which accept proposals of micro-nodes of u.letc = w p be the number of micro-receiers
Distributed Weighted Vertex Coer ia Maximal Matchings 845 of. The sum of the counter-proposals for node then needs to be at most c.at the same time each counter-proposal c u) must not exceed the corresponding proposal p u ). Gien these restrictions we choose a feasible set of counter-proposals {c u)} u N ) arbitrarily such that their sum is maximum, i.e. { u N ) c u)=min c, u N ) } p u ). Eentually, node decrements w by the sum of all the counter-proposals c u) and c u ) which hae been sent and receied by, respectiely: w = w u N ) c u)+c u )). This decrement reflects the number of micro-nodes of which are matched in the considered phase. If w becomes zero, node sends w to all its neighbors for the last time) and switches to the inside state since all the corresponding micro-nodes are matched). We now show how proposals are fixed by each node. If the number w of micronodes i) is sufficiently small, the proposals p u) are fixed according to algorithm M. Otherwise, they are fixed in a more efficient way, while keeping the same expected alue. In more details, there are two different strategies, depending on whether w < 2δ or not, where δ = N ) is the number of currently actie neighbors of.ifw < 2δ, the proposals p u) are initially set to zero. Then, for w times, an actie neighbor u N ) is selected at random with probability proportional to w u, and the corresponding proposal p u) is incremented by one with probability one half. Note that each p u), considered separately, is the sum of w i.i.d. Bernoulli ariables Bw u/2w )): p u)= w i=1 w ) B u 2W. 1) Otherwise w 2δ ), the alue of each p u) is independently set to: p u)= w w u w 2W + B w u 2W w w ) u 2W. 2) Note that, in both cases, the sum p of the proposals is upper bounded by w.in the first case this is triially true. In the second case, this is a consequence of the small alue of δ : w p u) w ) u u N ) u N 2W ) + 1 = w 2 + δ w. Moreoer, in both cases E[p ]=w /2. The following technical property of proposals will be useful in later parts of the analysis. Lemma 3. For any two gien nodes and u N ) we hae E u, = E [ 1 1 w ) ] pu ) 1 1 ) w u /4 W u.
846 Fabrizio Grandoni, Jochen Könemann, and Alessandro Panconesi Proof. If w u < 2δ u, by Equation 1): E u, = E 1 1 ) w ) u i=1 B w ) w = E 1 1 ) w w u B w = = 1 w + w 1 1 )) w u w 1 1 ) w u /2 W u. Consider now the case w u 2δ u. By Equation 2), if w u w / ) 1: [ E u, E 1 1 ) ] w u w /2W u) w 1 1 ) w u w /4W u) w 1 1 ) w u /4 W u. Otherwise w uw /) < 1): ) E u, = E 1 1 ) w B uw w = 1 w u 1 1 ) w u /2 W u. Lemma 4. Algorithm A computes a 2-approximate ertex coer. Proof. The algorithm halts. In fact, the residual weight of each actie node decreases by at least one in each round with positie probability. It follows that the nodes which do not switch to the outside state, switch to the inside state in a finite expected number of rounds. Assume by contradiction that, at the end of the algorithm, the inside nodes do not form a ertex coer. This implies that there is an outside node which has at least one outside neighbor. Let switch to the state outside in phase p. At the beginning of phase p 1), all the neighbors of are either inside or actie nodes. Consider the actie neighbors of in phase p 1). These nodes are not actie any more when phase p starts. But they cannot switch to the state outside in phase p 1), since their actie degree is greater than zero in that phase. Thus they all switch to the state inside,whichis a contradiction. Let z be the difference between w and the final residual weight w.a feasible solution of D) is obtained by assigning to each dual ariable y u the sum of all the counter-proposals of the kind c u) and c u ).Letapx and opt be the weight of the ertex coer found and that of a minimum ertex coer, respectiely. By weak duality: apx V z 2 u E y u 2opt. Thus the ertex coer found is 2-approximate. Lemma 5. Algorithm A sends messages of size OlogW). Each phase of algorithm A has time complexity O log W)) in expectation. Proof. Both proposals and counter-proposals can be packed in messages of size OlogW). The time complexity of each phase is upper bounded by the cost of computing the proposals. Computing the proposals is as expensie as selecting O ) times an element out of O W) ones uniformly at random. Each random selection can be
Distributed Weighted Vertex Coer ia Maximal Matchings 847 performed by generating Olog W)) random bits in expectation. By assuming a O1) cost for generating a random bit, the total expected cost of each phase is O log W)). Recall that a node is good if at least one third of its neighbors hae degree smaller or equal than its own degree. Consider a node in G. The degree of all the micro-nodes corresponding to in G is W = u N) w u. Thus a micro-node i) is good if and only if: w u W 3. u N):W u W Note that, if a micro-node i) is good, all the micro-nodes j), j {1,2...w },are good and ice-ersa. We call a node of G heay if all its micro-nodes in G are good. The next obseration can be seen as the weighted analogue of Lemma 1. Lemma 6. Let E H E be the subset of edges incident to heay nodes. Then u EH w w u 1 2 u E w w u. Proof. Consider the auxiliary graph G. The number of edges of G that are incident to good nodes is {,u} EH w w u. Since the number of edges of G is {,u} E w w u,the claim follows from Lemma 1. We use the properties of heay nodes to proe the following bound on the number of rounds. Lemma 7. Algorithm A halts in Ologn + logŵ ) expected rounds. Proof. We show that the residual weight of heay nodes decreases by at least a positie constant factor in expectation in each phase. It follows from Lemma 6 that the same holds for the potential function: 0 u E w w u < nŵ) 2, thus implying the claim. Consider a heay node in a gien phase. Let w be the alues of w at the end of the phase. The residual weight of decreases by at least the sum of the counter-proposals c u) sent by : ) w w c u) =p + c 1 1 c min{c, p u )} u N ) Note that: 1 1 c min{c, p u )} u N ) u N ) ) = p +w p ) 1 1 c min{c, p u )}. u N ) ) 1 1 ) u N ) p u ) c u N ) 1 1 ) pu ) w. Thus: [ E[w ] w 2 + w 2 E 1 1 ) ] pu ) u N w ),
848 Fabrizio Grandoni, Jochen Könemann, and Alessandro Panconesi whereweusede[p ]=w /2. By Lemma 3 and the definition of heay nodes: [ E 1 1 ) ] pu ) u N w ) 1 1 ) w u 4 u N W ) u u N ) W u W 1 1 ) w u 4 W 1 1 ) W 12. W The right-hand side is at most e 1/12 and it follows that E[w ] w 1 + e 1/12 )/2. Lemmas 4, 5, and 7 together imply Theorem 1. References 1. R. Bar-Yehuda and S. Een. A linear-time approximation algorithm for the weighted ertex coer problem. Journal of Algorithms, 2:198 203, 1981. 2. V. Chátal. Linear programming. Freeman, 1983. 3. T.H.Cormen,C.E.Leiserson,andR.L.Riest.Introduction to algorithms. MITPressand McGraw-Hill Book Company, 6th edition, 1992. 4. M. R. Garey and D. S. Johnson. Computers and Intractability. A Guide to the Theory of NP-Completeness. Freemann, 1979. 5. M. M. Halldórsson and J. Radhakrishnan. Greed is good: Approximating independent sets in sparse and bounded-degree graphs. In ACM Symposium on the Theory of Computing, pages 439 448, 23 25 1994. 6. E. Halperin. Improed approximation algorithms for the ertex coer problem in graphs and hypergraphs. SIAM Journal on Computing, 315):1608 1623, Oct. 2002. 7. M. Hańćkowiak, M. Karoński, and A. Panconesi. On the distributed complexity of computing maximal matchings. SIAM Journal on Discrete Mathematics, 151):41 57, 2001. 8. J. Håstad. Some optimal inapproximability results. Journal of the ACM, 484):798 859, July 2001. 9. D. S. Hochbaum. Efficient bounds for the stable set, ertex coer, and set packing problems. Discrete Applied Mathematics, 6:243 254, 1983. 10. A. Israeli and A. Itai. A fast and simple randomized parallel algorithm for maximal matching. Information Processing Letters, 22:77 80, 1986. 11. S. Khuller, U. Vishkin, and N. Young. A primal-dual parallel approximation technique applied to weighted set and ertex coer. Journal of Algorithms, 172):280 289, 1994. 12. B. Monien and E. Speckenmeyer. Ramsey numbers and an approximation algorithm for the ertex coer problem. Acta Informatica, 22:115 123, 1985. 13. A. Panconesi and R. Rizzi. Some simple distributed algorithms for sparse networks. DIST- COMP: Distributed Computing, 14, 2001. 14. C. Papadimitriou and M. Yannakakis. Optimization, approximization and complexity classes. Journal of Computer and System Sciences, 43:425 440, 1991.