R. Boppana and M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. BIT, 20:180--196, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....asymptotical inapproximability result (where the hardness factor is a function of the bound) and one for low values of the bound. 1.1 k Dimensional Matching The Unbounded Variant. The general problem of finding a matching in a hyper graph was extensively studied (for example [BYM84, BF94, BF95, BH92, Has99, Wig83] Quite tight approximation algorithms and inapproximability results are known for this problem. Hastad [Has99] proved that Set Packing cannot be approximated to within O(N ) unless NP ZPP (where N is the number of sets) The best approximation algorithm achieves an ....

....approximation algorithms and inapproximability results are known for this problem. Hastad [Has99] proved that Set Packing cannot be approximated to within O(N ) unless NP ZPP (where N is the number of sets) The best approximation algorithm achieves an approximation ratio of O( log ) BH92] In contrast, the case of bounded variants of this problem seems to be of a di#erent nature. Bounded Variants. The problem of finding a maximal matching in a bipartite graph (2 DM) is known to be solvable in polynomial time, say by a reduction to network flow problems [Pap94] Polynomial time ....

R. Boppana and Magnus M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. Bit 32, pages 180--196, 1992.

....to Cluster, and in fact, we have the same tractable intractable threshold as for the decision problem. The main open question is: how good is Maximum Cluster approximable depending on For instance, for (k) k 1 (i.e. Maximum Clique) it is known to be approximable within O (log n) [5] but not approximable within n unless P = NP [14] How do these results translate to intermediate densities Acknowledgment. For helpful hints and discussions we are grateful to Christian Gla er, Ernst W. Mayr, and Alexander O termatt Souza. We also thank Yuichi Asahiro, Refael Hassin, and ....

R. Boppana and M. M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32(2):180-196, 1992.

....cannot be computed exactly in polynomial time unless P = NP. For this reason, we can only hope to find e#cient algorithms that approximately solve the problem. The best known polynomial time approximation algorithm for MaxClique approximates within a ratio of (Boppana and Haldorsson [5]) A line of research, starting with Feige, Goldwasser, Lovasz, Safra, and Szegedy [9] and culminating in the work of Hastad [11] shows that we cannot expect to do much better: # This research was supported in part by National Science Foundation Grant 9988483. These ....

R. Boppana and M. M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32(2):180--196, 1992.

....ratio r together with a proof that, unless P = NP, there can be no approximation algorithm with performance ratio less than r. For the Maximum Independent Set problem, the best approximation algorithm is constructed by Boppana and Haldorsson, and has performance ratio O( V (log V ) BH92] On the other hand it is shown by Hastad that, unless NP = ZPP , it is impossible to approximate Maximum Independent Set within V for any # 0[Has99] In this chapter we will give an algorithm that approximates Matrix To Line# within 2. We will also show that, unless P = NP, the ....

Ravi Boppana and Magnus M. Halldorsson. Approximating Maximum Independent Sets by Excluding Subgraphs. BIT, 32:180--196, 1992.

....Science, Royal Institute of Technology, Stockholm, SWEDEN. turned to algorithms producing solutions which are at most some factor from the optimum value. It is trivial to approximate Max Clique in a graph with n vertices within n just pick any vertex as the clique and Boppana and Halldorsson [6] have shown that Max Clique can be approximated within O(n log n) in polynomial time. It is an astonishing, and unfortunate, result that it is hard to do substantially better than this. In fact, the Max Clique problem cannot be approximated within n , for any constant # 0, unless NP = ....

....of Samorodnitsky and Trevisan [22] we obtain the following concrete result regarding the approximability of Max Clique: Max Clique on a graph with n vertices cannot be approximated within n polynomial time. As a comparison, the best known polynomial time approximation algorithm [6], approximates Max Clique within n . Another problem akin to Max Clique is Min Chromatic Number, i.e. the problem of finding the minimum number of colors needed to properly vertex color a graph. In fact, results regarding the approximability of Min Chromatic Number are very similar to the ....

[Article contains additional citation context not shown here]

Ravi Boppana and Magnus M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. Bit, 32(2):180--196, June 1992.

....which are connected by an edge. For any constant k 3, if the given graph has an independent set of size n=k m, where n is the number of vertices, they obtain an Omega Gamma m k 1 log m) sized independent set, improving the previously known bound of k Gamma1 ) due to Boppana and Halldorsson[4]. All the new developments mentioned above are randomized algorithms. All of them share the following common paradigm. First, a semidefinite program is solved to obtain a collection of n vectors in n dimensional space satisfying some properties dependent upon the particular problem in question. ....

R. Boppana and M. Halldorsson, Approximating Maximum Independent Sets by Excluding Subgraphs, BIT, 32, pp. 180--196, 1992.

....107] R(Cm , K n ) m 1) n 1) 1 if m n except m = n = 3. 11 Approximation Algorithms The problem of finding an independent set of maximum size, or computing #(G) the independence number of a graph G, is one of the earliest problem shown to be NP hard. Boppana and Halldorsson (1992)[45], have a famous polynomial time approximation algorithm, based on Ramsey Theory, that finds an independent set of a guaranteed but not necessarily optimal size. The well known upperbound for o# diagonal Ramsey numbers stated in the classical Erdos Szekeres paper [96] gets a new algorithmic proof ....

....have a famous polynomial time approximation algorithm, based on Ramsey Theory, that finds an independent set of a guaranteed but not necessarily optimal size. The well known upperbound for o# diagonal Ramsey numbers stated in the classical Erdos Szekeres paper [96] gets a new algorithmic proof in [45] and it forms the basis for the algorithm, called Ramsey. The algorithm has the first non trivial performance guarantee for this problem. The performance guarantee is the largest ratio, over all inputs, of the size of the maximum independent set to the size of the approximation found. For the ....

[Article contains additional citation context not shown here]

R. Boppana and M.M. Halldorsson, Approximating maximum independent sets by excluding subgraphs, BIT 32 (

....f ] is defined analogously with f the amortized free bit complexity. Max Clique approximation. Although we look at many optimization problems there is a particular focus on Max Clique. Recall the best known polynomial time approximation algorithm for Max Clique achieves a factor of only N [BoHa], scarcely better than the trivial factor of N . Throughout the paper, when discussing the Max Clique problem, or any other problem about graphs, N denotes the number of vertices in the graph. Can one find even an N factor approximation algorithm for Max Clique for some ffl 1 An additional ....

....because of the cost of the reduction of [PaYa] which first reduces Max3SAT to its bounded version using expanders, and then reduces this to MinVC B. See Section 5.2 for our results. Max Clique. The best known polynomial time approximation algorithm for Max Clique achieves a factor of only N [BoHa], scarcely better than the trivial factor of N . There is not even a heuristic algorithm that is conjectured to do better. The Lov asz Theta function had been conjectured to approximate the Max Clique size within N but this conjecture was disproved by Feige [Fe1] Prior to 1991, no ....

R. Boppana and M. Hald' orsson. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992.

.... accepted by the on line algorithm is ON 2 ( i=1 OPT(I i ) 1) 2n ( i=1 OPT(I i ) OPT(I) 2n OPT: The current best known polynomial time algorithms have an approximation ratio of O( log n ) for the induced subgraph problem [Hal94] and O( log n) 2 ) for independent set [BH92] In particular, using the optimal (non polynomial time) algorithm for the problem (e.g. ff = n) we obtain: Corollary 12 There exists an O( n) competitive randomized algorithm for the on line induced subgraph problem. 4 On line edge disjoint paths In this section we present Omega Gamma ....

R.B. Boppana and M.M. Halldorson. Approximating maximum independent set by excluding subgraphs. BIT 32(2), pp.180-196, 1992.

....is always at most the size of the largest clique in G. We say that we have an f(n) approximation algorithm if this number is always at least the size of the largest clique divided by f(n) The best polynomial time approximation algorithm for MC achieves an approximation ratio of O( log n) 2 ) [12], and thus it is of the form n 1 o(1) This is not an easy result but note that an approximation factor of n is trivial since the clique cannot contain more than all n nodes and any set of a single node is a clique. On the negative side, there has been a sequence of papers [11, 17, 2, 1, 8, 18, ....

R. Boppana and M. Hald orsson. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992, pp 180-196.

....always at most the size of the largest clique in G. We say that we have an f(n) approximation algorithm if this number is always at least the size of the largest clique divided by f(n) The best polynomial time approximation algorithm for MC achieves an approximation ratio of O( n (log n) 2 ) [12], and thus it is of the form n 1 o(1) This is not an easy result but note that an approximation factor of n is trivial since the clique cannot contain more than all n nodes and any set of a single node is a clique. On the negative side, there has been a sequence of papers [11, 17, 2, 1, 8, 18, ....

R. Boppana and M. Hald orsson. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992, pp 180-196.

....this section shows that for any choice of a xed family H 0 of excluded r uniform (r 3) hypergraphs the local ratio approach cannot produce an approximation algorithm with approximation ratio asymptotically better than r. A result of a similar avor has been proven by Boppana and Halld orsson [5] for the case of graphs (r = 2) Let H 0 = fH 1 ; H t g be a xed family of r uniform hypergraphs. If we want to plug this family into our general algorithm LOCAL(H 0 ) and to obtain an algorithm with approximation ratio better than r, we should require that lr(H i ) r for every H i 2 H ....

....H 0 on n vertices, having the following properties: 1. H 0 does not contain a copy of any hypergraph from H 0 ; 2. H 0 does not contain an independent set of size dcn 1 (r 1) H 0 ) ln n) 1 r 1 e . Proof. This statement can be proven by applying the Lov asz local lemma [7] as it was done in [5]. We chose to present a proof based on using large deviation inequalities, as developed in [19] For every 1 i t let H 0 i be a subhypergraph of H i such that (H i ) jE(H 0 i )j 1) jV (H 0 i )j r) Setting H 0 0 = fH 0 1 ; H 0 t g, note that if H is H 0 0 free, then ....

R. Boppana and M. M. Halldorsson, Approximating maximum independent sets by excluding subgraphs, Bit 32 (1992), 180-196.

....is amongst the most important combinatorial optimization problems. Unfortunately it is NP hard [16] and attention since this discovery has thus focused on approximation algorithms. Yet the best known ones can approximate the max clique size of an N node graph only to within a factor of N 1 o(1) [11], scarcely better than the trivial factor of N . The rst twenty years following the NP hardness discovery brought little understanding of why this is so. Today we know a lot more. The results of [12, 3, 2] indicated the existence of a constant 0 for which N factor approximations are ....

R. Boppana and M. Hald orsson. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992.

....possible that they are even harder to approximate. Some problems, such as maximum clique and chromatic number, cannot be approximated [Has99, FK98] within n 1 Gammaffl , for any fixed ffl 0, unless P = ZPP. Known algorithms achieve approximation ratios of O(n= log 2 n) for maximum clique [BH92] and O(n(log log n) 2 = log 3 n) for chromatic number [Hal93] 1.3 Analysis of heuristics Although hard to solve in the worst case, NP hard problems may be significantly easier on average instances encountered in practice. It is therefore desirable to devise heuristic algorithms, that ....

.... r(n) It is known through work culminating in [Has99] that for any fixed ffl 0 it is impossible to approximate the clique number (G) within a ratio of n 1 Gammaffl , unless NP has randomized polynomial time algorithms (NP=ZPP) The best approximation algorithm that is known for (G) due to [BH92] has approximation ratio O(n= log 2 n) The intractability of the maximum independent set problem in the worst case suggests This chapter is based on [FK00a] and on [FK01b] 49 studying the performance of algorithms on average instances. A possible rigorous description of average instances ....

R. Boppana and M. M. Halld'orsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32:180--196, 1992.

....S is a packing of cuts. The claim follows. 2 While Theorem 2.1 implies that Cut Packing cannot be approximated better than Independent Set, Theorem 2.6 shows that any approximation guarantee for Independent Set immediately extends to Cut Packing by loosing a factor 2. Along with the results in [6], 5] 4] and [15] this leads to the following. 5 Corollary 2.7 The following approximation algorithms exist for Cut Packing (i) an O( n log 2 n ) approximation algorithm for general graphs; ii) a 2d 6 5 approximation algorithm for graphs with maximum degree d; iii) a (2 ....

R. Boppana, M.M. Halldorsson, Approximating Maximum Independent Sets by Excluding Subgraphs. Bit 32 (1992) 180-196.

.... number (G) within a factor O( np) 1=2 = log n) and having polynomial expected running time over G(n; p) Thus, in the most basic case p = 1=2 we get approximation algorithms with approximation ratio O(n 1=2 = log n) a considerable improvement over best known algorithms for the worst case [7], 13] whose approximation ratio is only O(n=polylog(n) Note also that the smaller the edge probability p(n) the better the approximation ratio is in both our results. Before turning to descriptions of our algorithms, we would like to say a few words about combinatorial ideas forming the ....

R. Boppana and M. M. Halldorsson, Approximating maximum independent sets by excluding subgraphs, Bit 32 (1992), 180-196.

....determining or estimating w(G) and that of nding a clique of maximum size in G are fundamental problems in Theoretical Computer Science. The problem of computing w(G) is well known to be NP hard [16] The best known approximation algorithm for this quantity, designed by Boppana and Halld orsson [8], has a performance guarantee of O(n= logn) 2 ) where n is the number of vertices in the graph. When the graph contains a large clique, there are better algorithms, and the best one, given in [3] shows that if w(G) exceeds n=k m, where k is a xed integer and m 0, then one can nd a clique ....

R. Boppana and M. M. Halldorsson, Approximating maximum independent sets by excluding subgraphs, BIT, 32 (1992),180-196.

....set of size (log k n) 2 . Finally, Halld orsson [11] came up with an approximation algorithm that uses at most (G)n(log log n) 2 = log n) 3 colors, currently best known result. His contribution is based on Ramsey type arguments for finding a large independent set from his paper with Boppana [6]. Both papers [3] and [11] proceed by repeatedly finding a large independent set, coloring it by a fresh color and discharging it quite a common approach in graph coloring algorithms. We will also adopt this strategy. It is worth noting here that one cannot hope for a major breakthrough in this ....

....do not seem to be applicable to the hypergraph case (i.e. when r 3) It is not clear how to define a notion of the neighborhood of a subset in order to apply the BergerRompel approach. Also, bounds on the hypergraph Ramsey numbers are too weak to lead to algorithmic applications in the spirit of [6], 11] However, something from the graph case can still be rescued. Both papers [9] and [15] dealing with the case of 2 colorable hypergraphs, noticed that the main idea behind Wigderson s algorithm is still usable for the hypergraph case. Let us describe now the main instrument of these papers, ....

[Article contains additional citation context not shown here]

R. Boppana and M. M. Halld'orsson, Approximating maximum independent sets by excluding subgraphs, Bit 32 (1992), 180--196.

....discuss an algorithm for the maximum independent set problem with a promise. We will assume that an input hypergraph H on n vertices contains an independent set of size n. The performance of our algorithm depends on . The graph version of this problem has been tackled by Boppana and Halld orsson [4] using the subgraph exclusion argument, and then by Alon and Kahale [1] based on the Lov asz function. Theorem 2.1. Let H be a 3 uniform hypergraph on n vertices, m edges and with an independent set of size at least n, for some constant 0. There exists a polynomial time algorithms which ....

R. Boppana and M. M. Halldorsson, Approximating maximum independent sets by excluding subgraphs, BIT 32 (1992), 180-196.

.... NP Max E3 SAT 1 1 7 folklore 1 1 37 unspecified [ALMSS] P 6= NP Max 2 SAT 1:075 [GoWi2, FeGo] 1:010 1 1 504 (implied [BeSu] P 6= NP MAX CUT 1:139 [GoWi2] 1:012 unspecified [ALMSS] P 6= NP Min VC 2 o(1) BaEv, MoSp] 1 1 26 unspecified [ALMSS] P 6= NP Max Clique N 1 o(1) [BoHa] N 1 4 [BeSu] NP 6 coR P N 1 3 coRP 6= NP N 1 4 N 1 5 [BeSu] P 6= NP Chromatic Num 5 N 1 o(1) BoHa] N 1 10 [BeSu] NP 6 coR P N 1 5 coRP 6= NP N 1 7 N 1 13 [BeSu] P 6= NP Figure 1: Approximation factors attainable by polynomial time ....

.... [BeSu] P 6= NP MAX CUT 1:139 [GoWi2] 1:012 unspecified [ALMSS] P 6= NP Min VC 2 o(1) BaEv, MoSp] 1 1 26 unspecified [ALMSS] P 6= NP Max Clique N 1 o(1) BoHa] N 1 4 [BeSu] NP 6 coR P N 1 3 coRP 6= NP N 1 4 N 1 5 [BeSu] P 6= NP Chromatic Num 5 N 1 o(1) [BoHa] N 1 10 [BeSu] NP 6 coR P N 1 5 coRP 6= NP N 1 7 N 1 13 [BeSu] P 6= NP Figure 1: Approximation factors attainable by polynomial time algorithms (Approx) versus factors we show are hard to achieve (Non Approx) Here 0 is an arbitrary positive constant. 1.2 Results for ....

R. BOPPANA AND M. HALD ORSSON. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992.

....determining or estimating w(G) and that of finding a clique of maximum size in G are fundamental problems in Theoretical Computer Science. The problem of computing w(G) is well known to be NPhard [14] The best known approximation algorithm for this quantity, designed by Boppana and Halld orsson [8], has a performance guarantee of O(n= logn) 2 ) where n is the number of vertices in the graph. When the graph contains a large clique, there are better algorithms, and the best one, given in [3] shows that if w(G) exceeds n=k m, where k is a fixed integer and m 0, then one can find a ....

R. Boppana and M. M. Halld'orsson, Approximating maximum independent sets by excluding subgraphs, BIT, 32 (1992),180--196.

....zero. Alon, Kahale and Szegedy [5] have also been able to use the semidefinite programming technique in conjunction with our techniques to obtain algorithms for computing bounds on the clique number of a graph with linearsized cliques, improving upon some results due to Boppana and Halldorsson [10]. In terms of disproving such a conjecture (or, proving upper bounds on and 0 ) relevant results include the following: Lovasz [32] points out that for a random graph G, G) n= log n while #(G) p n; Koniagin has demonstrated the existence of a graph which has (G) n=2 and #(G) ....

R.B. Boppana and M.M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32:180--196, 1992.

....of n 1 unless NP = ZPP . If a graph on n vertices is 3 colorable, then one can color it using O(n 3=14 log O(1) n) colors [5] but it is NP hard to color it using four colors [17] As for approximating the independence number of a graph, Boppana and Halld orsson presented an algorithm ([6]) with approximation ratio O(n= log 2 n) for graphs on n vertices, based on the so called Local Ratio Approach, to be discussed later in this paper. If a graph contains an independent set of size n m, for a constant 0 1=2, then an independent set of size O(m 3 1 ) can be found in ....

....independent set in a 3 uniform hypergraph. We will assume that an input hypergraph H on n vertices contains an independent set of size n, where 0 1 is a constant. The performance of our algorithm depends on . The graph version of this problem has been tackled by Boppana and Halld orsson [6] using the subgraph exclusion argument, and then by Alon and Kahale [1] based on the Lov asz function. We obtain the following result: Theorem 1 Let H be a 3 uniform hypergraph on n vertices, m edges and with an independent set of size at least n, for some constant 0. There exists a ....

[Article contains additional citation context not shown here]

R. Boppana and M. M. Halldorsson, Approximating maximum independent sets by excluding subgraphs, BIT 32 (1992), 180-196.

....vertex cover is equivalent to maximum independent set since the complement of a vertex cover is an independent set. However, for maximum independent set (or the equivalent maximum clique which is defined on the complement graph) the best known approximation factor is O(jV j= log jV j) 2 ) [10]. On the other hand, letting the worst independent set solution consist of the empty set, with zero weight, the z measure is exactly the ratio measure. Due to Hastad s hardness result [21] it follows that the problem is also hard to z approximate. Since z approximation is invariant under the ....

R. Boppana and M. M. Halld'orsson, "Approximating maximum independent sets by excluding subgraphs", BIT 32:180--196 (1992).

....vertex cover is equivalent to maximum independent set since the complement of a vertex cover is an independent set. However, for maximum independent set (or the equivalent maximum clique which is de ned on the complement graph) the best known approximation factor is O(jV j= log jV j) 2 ) [10]. On the other hand, letting the worst independent set solution consist of the empty set, with zero weight, the z measure is exactly the ratio measure. Due to Hastad s hardness result [21] it follows that the problem is also hard to z approximate. Since z approximation is invariant under the ....

R. Boppana and M. M. Halldorsson, \Approximating maximum independent sets by excluding subgraphs", BIT 32:180-196 (1992).

....An independent set in a graph G = V; E) is a set of pairwise non adjacent nodes. The Maximum Independent Set problem consists of finding an independent set of the largest cardinality. This problem is known to be np hard [9] to be approximable within factor O Gamma jV j= log jV j) 2 Delta [5], and to be not approximable within factor jV j 1 Gammaffl for any ffl 0 [12] unless corp = np) A vertex cover in a graph G = V; E) is a set of nodes such that each edge has at least one endpoint in the cover. The Minimum Vertex Cover problem consists of finding a cover of the smallest ....

Boppana, R., and Halld' orsson, M. M. Approximating maximum independent sets by excluding subgraphs. Bit 32 (1992), 180--196.

....size clique has at least a good polynomial time approximation algorithm. For the clique number, the best known performance guarantee of a polynomial time approximation algorithm is achieved by an algorithm due to Boppana and Halld orsson. It has a performance guarantee of O(n= log 2 n) [13]. Before the NP = PCP(log n; 1) result (Theorem 2.2) was proved, no non trivial lower bound for the performance guarantee of a polynomial time approximation algorithm for the clique number of a graph was known. The only result in this direction, due to Garey and Johnson, is that the existence of ....

R. Boppana and M.M. Halld'orsson (1992): Approximating maximum independent set by excluding subgraphs, BIT 32, 180--196, 1992.

....formulation of the problem has been used by Nemhauser and Trotter [30] In some cases, it is not known whether the same performance ratio is obtainable. For instance, while the unweighted Max Clique problem is approximable within O(n log 2 n) where n denotes the number of vertices in the graph [8], the best approximation algorithm for the weighted version of this problem reaches a factor of O( log log n) 2 n log 2 n) 20] Finally, it is well known that several NP hard optimization problems turn out to be tractable whenever a polynomial bound is imposed on the weights that appear in ....

R. Boppana and M.M. Halldorsson. Approximating maximum independent sets by excluding subgraphs, Bit 32:180--196, 1992.

....[22] For any positive integer k, Max EkSat = Max CSP(fOR k;j j0 j kg) The problem Max EkSat is a variant of Max kSat restricted to have clauses of length exactly k. Max Clique = Max Ones(NAND) Max Clique is known to be approximable to within a factor of O(n=log 2 n) in an n vertex graph [9] and is known to be hard to approximate to within a factor of Omega Gamma n 1 Gammaffl ) for any ffl 0 unless NP = RP [18, 23] We now go on to the minimization problems. The well known minimum s t cut problem in directed graphs is equivalent to Weighted Min CSP(F) for F = fOR 2;1 ; T ; Fg. ....

R. Boppana and M. Hald' orsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32(2), 180--196, 1992.

.... of O(n ) where is a xed positive constant de ned for MIS, 10, 2, 1] Under stronger complexity assumptions, MIS cannot be approximated to within a factor of O(n 0:5 ) 13] The best approximation factor for MIS found so far is a mere O( n log 2 n ) for the cardinality version, [5]. 2.1 Ramsey Theory and the Integrality Gap We just saw that one cannot expect to ever be able to nd a reasonable approximation for MIS. A tacit polyhedral combinatorics axiom is that if there is a reasonable approximation for MIS, there should be an LP relaxation for MIS which is solvable in ....

R. B. Boppana and M. M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32(2):180-196, 1992.

.... to nding the largest clique in the complementary graph, and cannot be approximated within a factor jV j 1 2 for any 0 unless P = NP , or within a factor jV j 1 for any 0 unless NP = ZPP [7] The best known approximation guarantee for this problem is O jV j (log jV j) 2 [3]. Conic programming We de ne the following convex cones: The n n symmetric matrices: Sn = X 2 IR n IR n ; X = X T ; The n n symmetric positive semide nite matrices: S n = X 2 Sn ; y T Xy 0 8y 2 IR n ; The n n symmetric copositive matrices: Cn = ....

R. Boppana and M.M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. Bit, 32:180-196, 1992.

....Royal Institute of Technology, Stockholm, SWEDEN. 1 turned to algorithms producing solutions which are at most some factor from the optimum value. It is trivial to approximate Max Clique in a graph with n vertices within n just pick any vertex as the clique and Boppana and Halldorsson [6] have shown that Max Clique can be approximated within O(n log 2 n) in polynomial time. It is an astonishing, and unfortunate, result that it is hard to do substantially better than this. In fact, the Max Clique problem cannot be approximated within n 1 # , for any constant # 0, unless NP = ....

....regarding the approximability of Max Clique: Theorem 8.1. Unless NP # ZPTIME(2 O(log n(log log n) 3 2 ) Max Clique on a graph with n vertices cannot be approximated within n 1 O(1 # log log n) in polynomial time. As a comparison, the best known polynomial time approximation algorithm [6], approximates Max Clique within n 1 O(log log n log n) Another problem akin to Max Clique is Min Chromatic Number, i.e. the problem of finding the minimum number of colors needed to properly 2 vertex color a graph. In fact, results regarding the approximability of Min Chromatic Number ....

[Article contains additional citation context not shown here]

Ravi Boppana and Magnus M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. Bit, 32(2):180--196, June 1992.

.... Max Cut is also MAX SNP complete [39] and the best known approximation algorithm for this problem, due to [22] achieves a performance ratio of 1:14 1= 878 10 Max Clique = Max Ones(NAND) Max Clique is known to be approximable to within a factor of O(n= log 2 n) in an n vertex graph [9] and is known to be hard to approximate to within a factor of Omega Gamma n 1 Gammaffl ) for any ffl 0 unless NP = RP [18, 23] We now go on to the minimization problems. The well known minimum s t cut problem in directed graphs is equivalent to Weighted Min CSP(F) for F = fOR 2;1 ; T ; Fg. ....

R. Boppana and M. Hald' orsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32(2), 180--196, 1992.

....been given for why (or if) this coding theoretic approach is a natural way to prove inapproximability results. We provide the following justification: in Problem Performance of best Best Hardness Limits of Code like poly time algorithm Result Reductions (this paper) Clique O(n= log 2 n) BH92] n 1=3 Gammaffl [BGS95] n 1=2 ffl Chromatic Number O(n= log 3 n) BH92] n 1=5 Gammaffl [F 94] n 1=2 ffl Coloring 3 color Uses O(n 0:25 ) 5 colors Cannot prove O(log n) able graphs colors [KMS94] required colors are required Vertex Cover 2 Gamma o(1) Hoc82] 1:01 ( PY91, BGS95] ....

....way to prove inapproximability results. We provide the following justification: in Problem Performance of best Best Hardness Limits of Code like poly time algorithm Result Reductions (this paper) Clique O(n= log 2 n) BH92] n 1=3 Gammaffl [BGS95] n 1=2 ffl Chromatic Number O(n= log 3 n) BH92] n 1=5 Gammaffl [F 94] n 1=2 ffl Coloring 3 color Uses O(n 0:25 ) 5 colors Cannot prove O(log n) able graphs colors [KMS94] required colors are required Vertex Cover 2 Gamma o(1) Hoc82] 1:01 ( PY91, BGS95] 1:5 ffl Table 1: Summary of known approximation properties of some ....

R. Boppana and M. Halld'orsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32:180--196, 1992.

....a partition of the graph into independent sets, the problems of approximating independent set and coloring are closely related. The dual problem to graph coloring is finding a clique cover, which is a partition of the graph into disjoint cliques. A preliminary version of this paper appeared in [9]. Supported in part by National Science Foundation Grant CCR 8902522 and PYI Award CCR 9057488. Researched done at Rutgers University. Supported in part by Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) fellowship. The analysis of approximation algorithms for ....

R. B. Boppana and M. M. Halld'orsson. Approximating maximum independent sets by excluding subgraphs. In Proc. of 2nd Scand. Workshop on Algorithm Theory. Lecture Notes in Computer Science #447, pages 13--25. Springer-Verlag, July 1990.

....are adjacent to at most a given number k vertices inside. We obtain performance ratios of O( # n) for the maximization versions of these problems. This is significantly better than what is known for the ordinary Independent Set problem, where the best performance ratio known is O(n log 2 n) [1], a mere log 2 n factor from trivial. In fact, it is known that obtaining a performance ratio that is any fixed root of n factor better than trivial is highly unlikely [4] We find that the same algorithmic technique extends to a number of related independence problems, for which no non trivial ....

..... In fact, no sub linear performance ratio is known for this problem. The maximization problem with # = 0 is trivial, whose solution consists of all isolated vertices. When # = N we have the Maximum Independent Set problem, for which the best performance ratio known is O(n log 2 n) [1]. Hastad has recently improved a sequence of deep results to show that this problem is hard to approximate within n 1 # , for any # 0 [4] This result is modulo the assumption that NP #= ZPP, namely that zero error randomized polynomial algorithms do not exist for all problems in NP . This ....

R. B. Boppana and M.M. Halldorsson, Approximating maximum independent sets by excluding subgraphs, BIT, 32 (1992), 180--196.

....coSMC and O(ae log min(n; p) for npSMC. Applying known results on the solvability and the approximability of WIS, we get for the pSMC approximation ratio of 16 on perfect graphs [GLS88] O( Delta log log Delta= log Delta) on boundeddegree graphs [H99] and O(n= log 2 n) on arbitrary graphs [BH92, H99] 6 Important special classes of graphs: We further study special classes of graphs. For pSMC, we describe an algorithm with a ratio of 1:5 for bipartite graphs. We generalize this to an algorithm with a ratio of (k 1) 2 on k colorable graphs, when the coloring is given. This, for ....

R. B. Boppana and M. M. Halld'orsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32:180--196, 1992.

No context found.

R. Boppana and M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. BIT, 20:180--196, 1992.

No context found.

R. Boppana and M. M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32(2):180--196, 1992.

No context found.

Ravi Boppana and Magnus M. Halldorsson. "Approximating Maximum Independent Sets by Excluding Subgraphs". In SWAT 90 2nd Scandinavian Workshop on Algorithm Theory, volume 447, pages 13--25, 1990.

No context found.

R. Boppana, M. Halldorsson. "Approximating maximum independent sets by excluding subgraphs ". BIT 32 (1992), 180--196.

No context found.

Ravi Boppana and Magnus M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. Bit, 32:180--196, 1992.

No context found.

R. Boppana and M. Hald' orsson. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992.

No context found.

R. Boppana and M. Hald'orsson, Approximating maximum independent sets by excluding subgraphs, BIT, 32 (1992), pp 180-196.

No context found.

R. Boppana and M. Hald' orsson. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992.

No context found.

R. Boppana and M. M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. Proc. 2nd Scandinavian Workshop on Algorithm Theory, 1990, 13--25.

No context found.

R. Boppana and M. Hald orsson, Approximating maximum independent sets by excluding subgraphs, BIT, 32 (1992), pp. 180--196.

No context found.

R. Boppana and M. Hald' orsson. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992.

No context found.

R. Boppana and M. Hald' orsson. Approximating maximum independent sets by excluding subgraphs. BIT, Vol. 32, No. 2, 1992.

No context found.

R. B. Boppana and M. M. Halldorsson. Approximating maximum independent sets by excluding subgraphs. BIT, 32(2):180--196, June 1992.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC