Distributed Weighted Vertex Cover via Maximal Matchings

Transcription

1 Distribted Weighted Vertex Coer ia Maximal Matchings FABRIZIO GRANDONI Uniersità di Roma Tor Vergata JOCHEN KÖNEMANN Uniersity of Waterloo and ALESSANDRO PANCONESI Sapienza Uniersità di Roma In this paper e consider the problem of compting a minimm-eight ertex-coer in an n-node, eighted, ndirected graph G = V, E). We present a flly distribted algorithm for compting ertex coers of eight at most tice the optimm, in the case of integer eights. Or algorithm rns in an expected nmber of Olog n + log Ŵ ) commnication ronds, here Ŵ is the aerage ertex-eight. The preios best algorithm for this problem reqires Olog nlog n + log Ŵ )) ronds and it is not flly distribted. For a maximal matching M in G it is a ell-knon fact that any ertex-coer in G needs to hae at least M ertices. Or algorithm is based on a generalization of this combinatorial loer-bond to the eighted setting. Categories and Sbject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnmerical Algorithms and Problems General Terms: Algorithms, Theory Additional Key Words and Phrases: Approximation algorithms, distribted algorithms, maximal matching, ertex coer 1. INTRODUCTION We are gien an ndirected graph G = V, E) and non-negatie integer ertex eights for all ertices V. A ertex coer is a sbset C V sch that each edge e E has at least one end-point in C. In the minimm-eight ertex-coer problem e ant to compte a ertex-coer of smallest total eight. A preliminary ersion of this paper appeared in COCOON 05. This ork as carried ot hile the first to athors ere at Sapienza Uniersità di Roma. Athors addresses: F. Grandoni, Dipartimento di Informatica, Sistemi e Prodzione, Uniersità di Roma Tor Vergata, Via del Politecnico 1, 00133, Roma, Italy, grandoni@disp.niroma2.it; J. Könemann, Department of Combinatorics and Optimization, Uniersity of Waterloo, 200 Uniersity Aene West, Waterloo, ON N2L 3G1, Canada, jochen@aterloo.ca; A. Panconesi, Dipartimento di Informatica, Sapienza Uniersità di Roma, Via Salaria 113, 00198, Roma, Italy, ale@di.niroma1.it. Permission to make digital/hard copy of all or part of this material ithot fee for personal or classroom se proided that the copies are not made or distribted for profit or commercial adantage, the ACM copyright/serer notice, the title of the pblication, and its date appear, and notice is gien that copying is by permission of the ACM, Inc. To copy otherise, to repblish, to post on serers, or to redistribte to lists reqires prior specific permission and/or a fee. c 20YY ACM /20YY/ $5.00 ACM Jornal Name, Vol. V, No. N, Month 20YY, Pages 1 13.

2 2 Fabrizio Grandoni et al. Compting minimm-eight ertex-coers is NP-hard [Garey and Johnson 1979]. Papadimitrio and Yannakakis [1991] sho that the problem is APX-hard. Recently, Dinr and Safra [2002] shoed that it is NP -hard to approximate the ertex-coer problem to ithin any factor smaller than > 1.36, improing on the preios best 7/6 loer bond by Håstad [2001]. On the positie side, the best knon approximation algorithm for the ertexcoer problem is de to Karakostas [2005] ho presented a 2 Θ1/ logn)) approximation for the problem. This improes pon earlier 2 o1)) approximation algorithms de to Bar-Yehda and Een [1981], Hochbam [1982], Monien and Speckenmeyer [1985], and Halperin [2002]. In the distribted setting, it is knon ho to compte a 2-approximate ertex coer in the neighted case. This can be achieed by compting a maximal matching in the graph and by inclding the matched nodes in the coer. A maximal matching can be compted in Olog 4 n) ronds ia the algorithm of Hanckoiack et al. [2001], and in O + log n) ronds ia the algorithm of Panconesi and Rizzi [2001]. Both algorithms are deterministic. Maximal matchings can also be compted in an expected nmber of Olog n) ronds ia the randomized algorithm of Israeli and Itai [1986]. For the eighted case a 2 + ɛ)-approximation can be compted deterministically in Olog n log 1 ɛ ) many ronds by sing the algorithm of Khller et al. [1994]. Their algorithm is stated as a PRAM algorithm, bt it is readily seen to be a bona fide distribted algorithm. Let Ŵ be the aerage eight. Then, by setting ɛ = 1/nŴ + 1), the latter algorithm comptes a 2-approximate ertex coer in Olog nlog n + log Ŵ )) commnication ronds. Note that the aboe choice of ɛ reqires global knoledge of the qantity nŵ. This assmption may not be realistic in all scenarios. In this paper e present an improed flly-distribted algorithm to compte a 2-approximate eighted ertex coer. Or main reslt can be stated as follos. Let W and denote the largest eight and degree of a node, respectiely. We recall that Ŵ is the aerage eight of a node. Theorem 1.1. There is a flly distribted algorithm hich comptes a 2-approximate eighted ertex coer in an expected nmber of Olog n + log Ŵ ) commnication ronds. The message size is Olog W ) and the local comptation done in each rond is O log W )) in expectation. For a maximal matching M in G it is a ell-knon fact that any ertex coer in G needs to hae at least M ertices. Or algorithm is based on a generalization of this property to the eighted setting. The basic idea is to expand each node of eight into micro-nodes 1), 2)..., ), and connect each i) to eery j) heneer is an edge of the netork. Then a maximal matching in the axiliary graph is compted. The ertex coer is gien by the nodes for hich all corresponding micro-nodes are matched. If the maximal matching is compted ia the flly-distribted algorithm of Israeli and Itai, the algorithm halts in an expected nmber of Olog n + log Ŵ ) ronds. A naie implementation of the matching algorithm by Israeli and Itai leads to psedo-polynomial message and time complexity in each rond. The main insight

3 Distribted Weighted Vertex Coer ia Maximal Matchings 3 leading to the bonds on message-size and local comptation time in Theorem 1.1 is to keep an implicit representation of the axiliary graph and a maximal matching in it. The rest of this paper is organized as follos. In Section 2 e introdce some preliminaries. Or algorithm relies on a carefl adaptation of the matching algorithm by Israeli and Itai. We present this adaptation in Section 3. Finally, Sections 4 and 5 deal ith the naie and refined ersions of or eighted ertex coer algorithm, respectiely. 2. PRELIMINARIES The minimm-eight ertex coer problem can be formlated as an integer linear program ILP): min V x s.t. x + x 1, E; x {0, 1}, V. Each assignment of the ariables hich satisfies the constraints feasible soltion) corresponds to a ertex coer containing exactly the nodes ith x = 1. By LP) e denote the natral linear programming relaxation of ILP). Let N) be the set of neighbors of. The linear programming dal D) of LP) is: max E y s.t. N) y, V ; y 0, E. By eak dality e.g., see [Chátal 1983]), the ale of each feasible soltion of D) is a loer bond for the ale of eery feasible soltion of LP) and hence ILP). In this paper e consider the standard synchronos message-passing model of comptation. The comptation proceeds in ronds. In each rond, a node can send/receie a message of nbonded size) to/from each one of its neighbors, and execte an nbonded amont of comptation. No global knoledge is aailable inclding the nmber n of nodes in the graph). The algorithms presented can be easily modified so as to ork in a non-falty) asynchronos system also. We se Bp), p [0, 1], to denote a Bernolli random ariable, hich takes ale 1 ith probability p and 0 otherise. A random bit is a Bernolli ariable B0.5). 3. DISTRIBUTED MAXIMAL MATCHING A matching of a graph G = V, E) is a sbset M E sch that no to edges of M are incident to the same node. The reslts of the next sections are based on the folloing simplified ersion M of the distribted maximal-matching algorithm of Israeli and Itai [Israeli and Itai 1986].

4 4 Fabrizio Grandoni et al. Algorithm M orks in phases, each one consisting of a constant nmber of ronds. In each phase, a matching is compted and the edges incident on matched nodes are remoed. The algorithm halts hen no edge is left. The maximal matching is gien by the nion of the matchings fond in the different phases. In a gien phase a matching is compted in the folloing ay. Let G = V, E ) be the crrent graph. By N ) and δ e denote the set of neighbors of and the degree of in G, respectiely. Each node randomly decides to be a sender or a receier ith probability one half. Note that the same node may play a different role in different phases. Each sender selects one neighbor N ) niformly at random and makes a proposal to. Each receier hich 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 hae degree δ δ. To proe the bond on the nmber of ronds, e se the folloing simple combinatorial reslt [Israeli and Itai 1986]: Lemma 3.1. At least one half of the edges of a graph are incident to good nodes. Theorem 3.2. Algorithm M comptes a maximal matching in Olog n) expected ronds. Proof. The correctness of the algorithm is triial. We sho that in each phase at least a constant fraction of the edges is remoed in expectation. This implies that the expected nmber of ronds is Ologn 2 )) = Olog n). Consider a good node of G in a gien phase. The probability P that accepts a proposal is loer bonded by: P From the definition of good nodes: N ) 1 1 ) 2δ N ):δ δ N ) 1 1 ) 2δ. 1 1 ) 2δ 1 1 ) δ 3 e 1 2δ 6. Ths P 1 e 1/6 )/2. Hence, by Lemma 3.1, the expected nmber of edges remoed is at least a fraction 1 e 1/6 )/4 of the total. 4. DISTRIBUTED VERTEX COVER VIA MAXIMAL MATCHINGS In this section e sho ho the problem of compting an approximate ertex coer can be redced to that of compting a maximal matching in an axiliary graph. Using this redction, e sho ho to compte a 2-approximate ertex coer in Olog n + log Ŵ ) expected ronds ia algorithm M of section 3. Consider the folloing axiliary graph G. For each node of G, G contains micro-nodes 1), 2)... ). To micro-nodes i) and j) are adjacent if and only if is an edge of G. In Figre 1 an example of the redction is gien. Let M be a maximal matching in G. By V M) e denote the set of nodes of G sch that all the corresponding micro-nodes i) are matched by M.

5 Distribted Weighted Vertex Coer ia Maximal Matchings Fig. 1. A eighted graph G on the left) ith the corresponding axiliary graph G. A maximal matching M of G is indicated ia dashed lines. The dashed nodes of G form a ertex coer. Lemma 4.1. Set V M) is a 2-approximate ertex coer of G. Proof. Assme by contradiction that V M) is not a ertex coer. Ths there are to adjacent nodes and in G hich do not belong to V M). This implies that there are to adjacent micro-nodes i) and j) in G hich are not matched by M. Then the set M = M {i)j)} is a matching, hich contradicts the maximality of M. Let apx and opt denote the eight of the ertex coer fond and that of a minimm eight ertex coer, respectiely. Moreoer, let z be the nmber of micro-nodes in {1), 2)... )} that are matched by M. A feasible soltion of D) is obtained by assigning to each dal ariable y the nmber of edges of the kind i)j) M. This soltion is feasible since, for eery V : y = z. By eak dality e obtain N) apx V and hence V M) is 2-approximate. z 2 E y 2 opt Lemma 4.1 sggests a strategy to compte a 2-approximate ertex coer distribtiely. The idea is to simlate the behaior of algorithm M on a irtal axiliary graph G, and then to select the nodes in the ertex coer as sggested by Lemma 4.1. Specifically, each node simlates the exection of the algorithm on the corresponding micro-nodes i) in G. Wheneer to micro-nodes i) and j) of G need to commnicate, nodes and are responsible for alloing sch commnication. The ertex coer is gien by the nodes sch that all the corresponding micro-nodes i) are matched by the maximal matching compted. Since the irtal axiliary graph contains OnŴ ) nodes, the total nmber of ronds is OlognŴ )) = Olog n + log Ŵ ). This naie application of Lemma 4.1 has to major drabacks. The first problem

6 6 Fabrizio Grandoni et al. is the large message size. In fact, in each phase all the remaining) micro-nodes of may send a proposal to some micro-node of. Ths 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, ith probability one half, needs to select one neighbor ot of Θ W ) niformly at random. This random selection can be performed in Θlog W )) expected time, assming that the cost of generating a random bit is O1) e.g., see [Cormen et al. 1992]). Ths the expected time complexity of each phase is ΩW log W )). In next section e sho ho to sole both problems by creating the matchings implicitly. 5. AN IMPROVED ALGORITHM In this section e present an improed flly distribted algorithm A for compting a 2-approximate ertex coer. Algorithm A still reqires Olog n+log Ŵ ) expected ronds, bt it redces the size of the messages to Olog W ) and the expected time complexity of each phase to O log W )). The basic strctre of algorithm A is analogos to the strctre of the naie algorithm described in preios section: in each phase, a matching in the crrent axiliary graph G is compted, and the matched nodes are remoed from G together ith all the edges incident to them). The algorithm halts hen no edge is left. The ertex coer is gien by the nodes sch that all the corresponding micro-nodes i) are matched by one of the matchings compted. The main noelty in Algorithm A is that matchings are created implicitly: in each phase each node only knos the nmber of the corresponding matched micronodes. Intitiely, this simplification is alloed by the symmetry properties of G: all the remaining micro-nodes corresponding to a node hae the same degree and share the same neighborhood. This inariant is kept by all the indced sbgraphs of G. Algorithm A, hich is described in Figre 2, orks in phases. Each phase consists of a constant nmber of commnication ronds. Each node has an associated state s, hich is initially actie. In each phase, some of the actie nodes sitch to the state inside or otside, and the algorithm terminates hen 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 residal eight, hich is initially. The residal eight can be interpreted as the nmber of micronodes i) of in the crrent axiliary graph G. Note that all the micro-nodes i) hae the same degree W : W =. N) In each phase, the expected residal eight of actie nodes decreases. The decrease of in a gien phase reflects the nmber of micro-nodes of that hae been matched in that phase. At the beginning of each phase, each actie node sends to all its crrently actie neighbors N ). The neighbors ith = 0 are remoed from N ). If

7 Distribted Weighted Vertex Coer ia Maximal Matchings 7 = ; N ) = N), s = actie; hile s = actie) { send and receie to/ from all N ); N ) = { N) : > 0}; if N ) = 0) s = otside; else { compte the proposals p ); send p ) and receie p ) to/ from all N ); compte the conter-proposals c ); send c ) and receie c ) to/ from all N ); for all N )) = c) c); if = 0) { send to all N ); s = inside; } } } Fig. 2. Protocol for node for 2-approximate ertex coer. N ) becomes empty, node sitches to the otside state. In fact, in this case the degree W of the micro-nodes i) is zero, and ths they ill neer be matched. Otherise, sends a proposal p ) to each actie neighbor N ). The ale of p ) can be interpreted as the nmber of proposals directed from the micro-nodes of to the micro-nodes of. Let p be the sm of the proposals p ): p = N ) p ). This qantity can be ieed as the nmber of micro-senders among 1), 2),..., ). We postpone a detailed description of ho proposals are fixed ntil later. For each proposal p ) receied, node replies ith a conter-proposal c ). The conter-proposal c ) can be interpreted as the nmber of micro-nodes of hich accept proposals of micro-nodes of. Let c = p be the nmber of micro-receiers of. The sm of the conter-proposals for node then needs to be at most c there cannot be more accepted proposals than micro-receiers in ). At the same time each conter-proposal c ) mst not exceed the corresponding proposal p ) micro-receiers of cannot accept more proposals from the microsenders of than the proposals actally receied). Gien these restrictions, e choose a feasible set of conter-proposals {c )} N ) arbitrarily, sch that their sm is maximm. For example, let N ) = { 1, 2,..., N ) }. For increasing ales of i, i = 1, 2,..., N ), e can set c i ) = min{p ), c i 1 j=1 c j )}. Obsere that, at the end of the process, N ) c ) = min{c, N ) p )}. Eentally, node decrements by the sm of all the conter-proposals c )

8 8 Fabrizio Grandoni et al. and c ) hich hae been sent and receied by, respectiely: = c ) + c )). N ) This decrement reflects the nmber of micro-nodes of hich are matched in the considered phase. If becomes zero, node sends to all its neighbors for the last time) and sitches to the inside state since all the corresponding micro-nodes are matched). Obsere that the ale of any feasible proposal and ths of any conter-proposal) is at most W. Hence, the message size decreases to Olog W ). It remains to sho ho to compte the proposals efficiently, ithot increasing the expected nmber of commnication ronds. A feasible set of proposals for a gien node is obiosly obtained by simlating Algorithm M: the proposals p ) are initially set to zero. Then, for times, an actie neighbor N ) is selected at random ith probability proportional to, and the corresponding proposal p ) is incremented by one ith probability one half. Note that each p ), considered separately, is the sm of i.i.d. Bernolli ariables B ): ) p ) = B. 1) i=1 Hoeer, this approach is too expensie for large ale of. The idea is then to approximate the behaior of the proposals aboe. A natral choice is as follos: p ) = + B ). 2) We obsere that, ith both 1) and 2), E[p )] =. Hoeer, the comptation of 2) is faster. There is a technical problem: for the proposals to be feasible, it mst be deterministically), p := N ) p ) there cannot be more micro-senders than micro-nodes). This is garanteed for 2δ, here δ = N ) 1 is the nmber of crrently actie neighbors of : p ) N ) N ) ) + 1 = 2 + δ. For < 2δ 2, e jst simlate Algorithm M as described before. Smmarizing, proposals are set in the folloing ay: ) i=1 p ) = B if < 2δ ) ; + B otherise. The folloing technical property of the proposals ill be sefl in later parts of the analysis.

9 Distribted Weighted Vertex Coer ia Maximal Matchings 9 Lemma 5.1. For fixed { } V, and for any to gien actie nodes and N ), [ E, := E 1 1 ) ] p) e 4W. Proof. Recall that, for x 1, 1 1 x )x e 1 and x x/2. We also remark that W 1. If < 2δ, ) ) E, = E 1 1 ) i=1 B = E 1 1 ) B = )) = 1 1 ) e. 1, Consider no the case 2δ. We frther distingish to sbcases. If E, = E 1 1 ) +B ) E 1 1 ) = 1 1 Otherise E, = E 1 1 ) 1 1 ) 4W < 1): ) +B ) e 4W. = E 1 1 ) ) B = ) = 1 e. Lemma 5.2. Algorithm A comptes a 2-approximate ertex coer. Proof. The algorithm halts. In fact, the residal eight of each actie node decreases by at least one in each rond ith positie probability. It follos that the nodes hich do not sitch to the otside state, sitch to the inside state in a finite expected nmber of ronds. Assme by contradiction that, at the end of the algorithm, the inside nodes do not form a ertex coer. This implies that there is an otside node hich has at least one otside neighbor. Let sitch to the state otside 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 hen phase p starts. Bt they cannot sitch to the state otside in phase p 1), since their actie degree is greater than zero in that phase. Ths they all sitch to the state inside, hich is a contradiction.

10 10 Fabrizio Grandoni et al. Let z be the difference beteen and the final residal eight. A feasible soltion of D) is obtained by assigning to each dal ariable y the sm of all the conter-proposals of the kind c ) and c ). Let apx and opt be the eight of the ertex coer fond and that of a minimm ertex coer, respectiely. By eak dality: apx z 2 y 2 opt. V E Ths the ertex coer fond is 2-approximate. Lemma 5.3. Algorithm A sends messages of size Olog W ). Each phase of algorithm A has time complexity O log W )) in expectation. Proof. Both proposals and conter-proposals can be packed in messages of size Olog W ). The time complexity of each phase is pper bonded by the cost of compting the proposals. Compting the proposals is as expensie as selecting O ) times an element ot of O W ) ones niformly at random. Each random selection can be performed by generating Olog W )) random bits in expectation. By assming 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 eqal than its on degree. Consider a node in G. The degree of all the micro-nodes corresponding to in G is: W =. N) Ths a micro-node i) is good if and only if: N):W W W 3. Note that, if a micro-node i) is good, all the micro-nodes j), j {1, 2... }, 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 eighted analoge of Lemma 3.1. Lemma 5.4. Let E H E be the sbset of edges incident to heay nodes. Then: 1. 2 E H E Proof. Consider the axiliary graph G. The nmber of edges of G that are incident to good nodes is: {,} E H. Since the nmber of edges of G is {,} E, the claim follos from Lemma 3.1. We se the properties of heay nodes to proe the folloing bond on the nmber of ronds. Lemma 5.5. Algorithm A halts in Olog n + log Ŵ ) expected ronds.

11 Distribted Weighted Vertex Coer ia Maximal Matchings 11 Proof. We sho that the residal eight of heay nodes decreases by at least a positie constant factor in expectation in each phase. It follos from Lemma 5.4 that the same holds for the potential fnction: 0 E < nŵ )2, ths implying the claim. We condition on the ales { } V at the beginning of a gien phase. Consider a heay node in that phase. Let be the ale of at the end of the phase. The residal eight of decreases by at least the sm of the conter-proposals c ) sent by : N ) c ) = min{c, here, by definition, = p + c. Triially, E[ ] = P rc = 0)E[ c = 0] + P rc 1)E N ) N ) = P rc = 0)E[p c = 0] + P rc 1)E p + c = E[p ] + P rc 1)E c No obsere that, for c 1, c N ) N ) c ) = c min{c, c ) c 1. N ) p )} p )}, c ) c 1 N ) = c 1 min{c, N ) p ) )} c = c {0, max N 1 ) p } ) c c 1 1 ) N ) p) c c 1 1 ) p). N ) c ) c 1 Recall that e condition oer the ales { } V. Under this condition, c and

12 12 Fabrizio Grandoni et al. the p ) s are independent. By this obseration and the ineqality aboe: E c c ) c 1 E c 1 1 ) p) c 1 N ) N ) [ = E[c c 1] E 1 1 ) ] p) c 1 N ) = E [c c 1] By Lemma 5.1 and the definition of heay nodes: [ E 1 1 ) ] p) e 4W N ) N ) We notice that E[p ] = E[c ] = /2. Moreoer Altogether N ) E [ 1 1 N ):W W E[c ] = P rc = 0)E[c c = 0] + P rc 1)E[c c 1] = P rc 1)E[c c 1]. E[ ] = E[p ] + P rc 1)E c N ) E[p ] + P rc 1)E[c c 1]e 1 12 c ) c 1 = E[p ] + E[c ]e 1 12 = e 1 12 <. ) p) ]. e 4W e Lemmas 5.2, 5.3, and 5.5 together imply Theorem 1.1. REFERENCES Bar-Yehda, R. and Een, S A linear-time approximation algorithm for the eighted ertex coer problem. Jornal of Algorithms 2, Chátal, V Linear programming. Freeman. Cormen, T. H., Leiserson, C. E., and Riest, R. L Introdction to algorithms, 6th ed. MIT Press and McGra-Hill Book Company. Dinr, I. and Safra, S The importance of being biased. In ACM Symposim on the Theory of Compting STOC) Garey, M. R. and Johnson, D. S Compters and Intractability. A Gide to the Theory of NP-Completeness. Freemann. Halperin, E Improed approximation algorithms for the ertex coer problem in graphs and hypergraphs. SIAM Jornal on Compting 31, 5, Hańćkoiak, M., Karoński, M., and Panconesi, A On the distribted complexity of compting maximal matchings. SIAM Jornal on Discrete Mathematics 15, 1, Håstad, J Some optimal inapproximability reslts. Jornal of the ACM 48, 4, Hochbam, D. S Approximation algorithms for set coering and ertex coer problems. SIAM Jornal on Compting 11,

13 Distribted Weighted Vertex Coer ia Maximal Matchings 13 Israeli, A. and Itai, A A fast and simple randomized parallel algorithm for maximal matching. Information Processing Letters 22, Karakostas, G A better approximation ratio for the ertex coer problem. In International Colloqim on Atomata, Langages and Programming ICALP) Khller, S., Vishkin, U., and Yong, N A primal-dal parallel approximation techniqe applied to eighted set and ertex coer. Jornal of Algorithms 17, 2, Monien, B. and Speckenmeyer, E Ramsey nmbers and an approximation algorithm for the ertex coer problem. Acta Informatica 22, Panconesi, A. and Rizzi, R Some simple distribted algorithms for sparse netorks. DISTCOMP: Distribted Compting 14. Papadimitrio, C. and Yannakakis, M Optimization, approximization and complexity classes. Jornal of Compter and System Sciences 43, Receied March 2006; reised December 2007; accepted May 2008.