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Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 958 (14 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...
Lowdegree Graph Partitioning via Local Search with Applications to Constraint Satisfaction, Max Cut, and Coloring
 JOURNAL OF GRAPH ALGORITHMS AND APPLICATIONS
, 1997
"... We present practical algorithms for constructing partitions of graphs into a fixed number of vertexdisjoint subgraphs that satisfy particular degree constraints. We use this in particular to find kcuts of graphs of maximum degree \Delta that cut at least a k\Gamma1 k (1 + 1 2\Delta+k\Gamma1 ) fr ..."
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Cited by 24 (6 self)
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We present practical algorithms for constructing partitions of graphs into a fixed number of vertexdisjoint subgraphs that satisfy particular degree constraints. We use this in particular to find kcuts of graphs of maximum degree \Delta that cut at least a k\Gamma1 k (1 + 1 2\Delta+k\Gamma1 ) fraction of the edges, improving previous bounds known. The partitions also apply to constraint networks, for which we give a tight analysis of natural local search heuristics for the maximum constraint satisfaction problem. These partitions also imply efficient approximations for several problems on weighted boundeddegree graphs. In particular, we improve the best performance ratio for the weighted independent set problem to 3 \Delta+2 , and obtain an efficient algorithm for coloring 3colorable graphs with at most 3\Delta+2 4 colors.
Fast Approximation Algorithms on Maxcut, kColoring and kColor Ordering for VLSI Applications
, 1994
"... There is a number of VLSI problems that have a common structure. We investigate such a structure that leads to a unified approach for three independent VLSI layout problems: partitioning, placement and via minimization. Along the line, we first propose a lineartime approximation algorithm on maxcut ..."
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Cited by 8 (0 self)
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There is a number of VLSI problems that have a common structure. We investigate such a structure that leads to a unified approach for three independent VLSI layout problems: partitioning, placement and via minimization. Along the line, we first propose a lineartime approximation algorithm on maxcut and two closely related problems: k coloring and maximal kcolor ordering problem. The kcoloring is a generalization of the maxcut and the maximal kcolor ordering is a generalization of the kcoloring. For a graph G with e edges and n vertices, our maxcut approximation algorithm runs in O(e + n) sequential time yielding a nodebalanced maxcut with size at least (w(E) + w(E)=n)=2, improving the time complexity of O(e log e) known before. Building on the proposed maxcut technique and employing a heightbalanced binary decomposition, we devise an O((e + n) log k) time algorithm for the kcoloring problem which always finds a kpartition of vertices such that the number of bad (or "defec...
Advanced Scatter Search for the MaxCut Problem
"... The MaxCut problem consists of finding a partition of the nodes of a weighted graph into two subsets such that the sum of the weights between both sets is maximized. This is an NPhard problem that can also be formulated as an integer quadratic program. Several solution methods have been developed ..."
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Cited by 6 (0 self)
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The MaxCut problem consists of finding a partition of the nodes of a weighted graph into two subsets such that the sum of the weights between both sets is maximized. This is an NPhard problem that can also be formulated as an integer quadratic program. Several solution methods have been developed since the 1970s and applied to a variety of fields, particularly in engineering and layout design. We propose a heuristic method — based on the scatter search methodology — for finding approximate solutions to this optimization problem. Our solution procedure incorporates some innovative features within the scatter search framework: 1) the solution of the maximum diversity problem to increase diversity in the reference set, 2) a dynamic adjustment of a key parameter within the search, and 3) the adaptive selection of a combination method. We perform extensive computational experiments to first study the effect of changes in critical scatter search elements and then to compare the efficiency of our proposal with previous solution procedures. Keywords: Maxcut problem, scatter search, metaheuristics, evolutionary algorithms Version: June 1, 2007Advanced Scatter Search for the MaxCut Problem / 2 1.
On Greedy Construction Heuristics for the MaxCut Problem
 INTERNATIONAL JOURNAL ON COMPUTATONAL SCIENCE AND ENGINEERING
, 2007
"... Given a graph with nonnegative edge weights, the MAXCUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in di#erent subsets is maximized. This classical NPhard problem finds applications in VLSI design, statistical physics, an ..."
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Cited by 5 (0 self)
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Given a graph with nonnegative edge weights, the MAXCUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in di#erent subsets is maximized. This classical NPhard problem finds applications in VLSI design, statistical physics, and classification among other fields. This paper compares the performance of several greedy construction heuristics for MAXCUT problem. In particular, a new "worstout" approach is studied and the proposed edge contraction heuristic is shown to have an approximation ratio of at least 1/3. The results of experimental comparison of the worstout approach, the wellknown bestin algorithm, and modifications for both are also included.
Parametrizing Above Guaranteed Values: MaxSat and MaxCut
, 1997
"... In this paper we investigate the parametrized complexity of the problems MaxSat and MaxCut using the framework developed by Downey and Fellows[7]. Let G be an arbitrary graph having n vertices and m edges, and let f be an arbitrary CNF formula with m clauses on n variables. We improve Cai and Chen's ..."
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Cited by 4 (0 self)
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In this paper we investigate the parametrized complexity of the problems MaxSat and MaxCut using the framework developed by Downey and Fellows[7]. Let G be an arbitrary graph having n vertices and m edges, and let f be an arbitrary CNF formula with m clauses on n variables. We improve Cai and Chen's O(2 2k m) time algorithm for determining if at least k clauses of of a cCNF formula f can be satisfied[4]; our algorithm runs in O(jf j +k 2 OE k ) time for arbitrary formulae and in O(m + kOE k ) time for cCNF formulae. We also give an algorithm for finding a cut of size at least k; our algorithm runs in O(m + n + k4 k ) time. Since it is known that G has a cut of size at least d m 2 e and that there exists an assignment to the variables of f that satisfies at least d m 2 e clauses of f , we argue that the standard parametrization of these problems is unsuitable. Nontrivial situations arise only for large parameter values, in which range the fixedparameter tractable a...
Approximating Maximum 2CNF Satisfiability
, 1992
"... A parallel approximation algorithm for the MAXIMUM 2CNF SATISFIABILITY problem is presented. This algorithm runs in O(log 2 (n + jF j)) parallel time on a CREW PRAM machine using O(n+jF j) processors, where n is the number of variables and jF j is the number of clauses. Performance guarantees are ..."
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Cited by 2 (0 self)
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A parallel approximation algorithm for the MAXIMUM 2CNF SATISFIABILITY problem is presented. This algorithm runs in O(log 2 (n + jF j)) parallel time on a CREW PRAM machine using O(n+jF j) processors, where n is the number of variables and jF j is the number of clauses. Performance guarantees are considered for three slightly differing definitions of this problem. Keywords : Satisfiability, Maximum 2CNF SAT, Maximum Cut, Approximation Algorithm. 1. Introduction A satisfiability problem takes as input a formula which is a conjunction of clauses F = (c 1 ; : : : ; c m ). Let jF j denote m, the number of clauses in F . Each clause c i is a disjunction of r i literals, where each literal is either a positive (true) or negative (false) appearance of a variable from the set X = fx 1 ; : : : ; xng. Such a boolean formula is said to be in conjunctive normal form (CNF). The objective is to find a truth assignment of the n variables that satisfies (makes the boolean clause true) either a...
Bounding Probability of Small Deviation: A Fourth Moment Approach
, 2007
"... In this paper we study the problem of bounding the value of the probability distribution function of a random variable X at E[X] + a where a is a small quantity in comparison with E[X], by means of the second and the fourth moments of X. In this particular context, many classical inequalities yield ..."
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Cited by 2 (0 self)
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In this paper we study the problem of bounding the value of the probability distribution function of a random variable X at E[X] + a where a is a small quantity in comparison with E[X], by means of the second and the fourth moments of X. In this particular context, many classical inequalities yield only trivial bounds. By studying the primaldual momentsgenerating conic optimization problems, we obtain upper bounds for Prob {X ≥ E[X] + a}, Prob {X ≥ 0}, and Prob {X ≥ a} respectively, where we assume the knowledge of the first, second and fourth moments of X. These bounds are proved to be tightest possible. As application, we demonstrate that the new probability bounds lead to a substantial sharpening and simplification of a recent result and its analysis by Feige ([7], 2006); also, they lead to new properties of the distribution of the cut values for the maxcut problem. We expect the new probability bounds to be useful in many other applications.
Efficient Approximation Algorithms for Some Semidefinite Programs
, 1996
"... ization problems. Nonlinear programming did not receive as much attention in this respect until the recent work by Goemans and Williamson [62]. They use semidefinite programs, which are nonlinear programs, to obtain approximation solutions with much better approximation factors. For example, the bes ..."
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Cited by 1 (0 self)
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ization problems. Nonlinear programming did not receive as much attention in this respect until the recent work by Goemans and Williamson [62]. They use semidefinite programs, which are nonlinear programs, to obtain approximation solutions with much better approximation factors. For example, the best previously known approximation algorithm for MAXCUT, which was invented twenty years ago, has approximation factor 0.5 [137]. The algorithm of Goemans and Williamson dramatically improves the approximation factor to 0.878. Inspired by the work on MAXCUT, Karger, Motwani, and Sudan [86] adapt the same technique and obtain the currently best approximation algorithm for coloring a kcolorable graph with the fewest possible number of colors. The approximation ratio is improved by a factor of \Omega\Gamma n 2=k ) over the best previously known result [29]. Later Karger and Blum give the best known approximation algorithm for color
New Approximate Graph Matching and Maximum cut
"... . For a graph G with e edges and n vertices, and w(E) as a total edge weight, a maximum cardinality matching (MCM) (resp. maximum weighted matching (MWM)) of G is a maximum subset M of edges (resp. a subset M of edges with a maximum edge weight) such that no two edges of M are incident at a common v ..."
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. For a graph G with e edges and n vertices, and w(E) as a total edge weight, a maximum cardinality matching (MCM) (resp. maximum weighted matching (MWM)) of G is a maximum subset M of edges (resp. a subset M of edges with a maximum edge weight) such that no two edges of M are incident at a common vertex. The best known algorithm for solving the MCM problem in general graphs (resp. the MWM problem in bipartite graphs) requires O(n 5=2 ) time (resp. O(n(e + n log n)) time). We first propose an approximate MCM algorithm that runs in O(e + n) sequential time yielding a matching of size at least e n\Gamma1 . Next, the proposed MCM algorithm is extended to the weighted case running in O(e + n) time, yielding the size of at least w(E) n\Gamma1 , when n is even. When n is odd, the lower bound obtained is w(E)\Gammaw(I v ) n\Gamma2 , where w(Iv) is the weight on edges incident to vertex v which is minimum over considering all vertices. The results improve the bound known before. The ...