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28
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 175 (26 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
A 7/8Approximation Algorithm for MAX 3SAT?
 IN PROCEEDINGS OF THE 38TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1997
"... We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instancea collection of clauses each of length at most threeis satisfiable, then the expected weight of the assignment found is at least 7=8 of optimal. We provide strong evidence (but not a p ..."
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Cited by 108 (10 self)
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We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instancea collection of clauses each of length at most threeis satisfiable, then the expected weight of the assignment found is at least 7=8 of optimal. We provide strong evidence (but not a proof) that the algorithm performs equally well on arbitrary MAX 3SAT instances. Our algorithm uses semidefinite programming and may be seen as a sequel to the MAXCUT algorithm of Goemans and Williamson and the MAX 2SAT algorithm of Feige and Goemans. Though the algorithm itself is fairly simple, its analysis is quite complicated as it involves the computation of volumes of spherical tetrahedra. Hastad has recently shown that, assuming P 6= NP , no polynomialtime algorithm for MAX 3SAT can achieve a performance ratio exceeding 7=8, even when restricted to satisfiable instances of the problem. Our algorithm is therefore optimal in this sense. We also describe a method of obtaining direct semi...
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 96 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Approximation Algorithms for Constraint Satisfaction Problems Involving at Most Three Variables per Constraint
 In Proceedings of the 9th Annual ACMSIAM Symposium on Discrete Algorithms
, 1997
"... An instance of MAX 3CSP is a collection of m clauses of the form f i (z i1 ; z i2 ; z i3 ), where the z ij 's are literals, or constants, from the set f0; 1; x 1 ; : : : ; xn ; x 1 ; : : : ; xng, and the f i 's are arbitrary Boolean functions depending on (at most) three variables. Each clause h ..."
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Cited by 83 (6 self)
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An instance of MAX 3CSP is a collection of m clauses of the form f i (z i1 ; z i2 ; z i3 ), where the z ij 's are literals, or constants, from the set f0; 1; x 1 ; : : : ; xn ; x 1 ; : : : ; xng, and the f i 's are arbitrary Boolean functions depending on (at most) three variables. Each clause has a nonnegative weight w i associated with it. A solution to the instance is an assignment of 01 values to the variables x 1 ; : : : ; xn that maximizes P n i=1 w i f i (z i1 ; z i2 ; z i3 ), the total weight of the satisfied clauses. The MAX 3CSP problem is clearly a generalization of the MAX 3SAT problem. (In an instance of the MAX 3SAT problem f i (z i1 ; z i2 ; z i3 ) = z i1 z i2 z i3 for every i.) Karloff and Zwick have recently obtained a 8 approximation algorithm for MAX 3SAT. Their algorithm is based on a new semidefinite relaxation of the problem. Hastad showed that no polynomial time algorithm can achieve a better performance ratio, unless P=NP. Here we use similar techniques to obtain a approximation algorithm for MAX 3CSP. The performance ratio of this algorithm is also optimal, as follows again from the work of Hastad. We also obtain better performance ratios for several special cases of the problem. Our results include: 2 approximation algorithm for MAX 3AND, the problem in which each clause is of the form z i1 z i2 z i3 . This result is optimal and it implies the result for MAX 3CSP.
Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems
, 1999
"... We present a tool, outward rotations, for enhancing the performance of several semidefinite programming based approximation algorithms. Using outward rotations, we obtain an approximation algorithm for MAX CUT that, in many interesting cases, performs better than the algorithm of Goemans and William ..."
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Cited by 60 (7 self)
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We present a tool, outward rotations, for enhancing the performance of several semidefinite programming based approximation algorithms. Using outward rotations, we obtain an approximation algorithm for MAX CUT that, in many interesting cases, performs better than the algorithm of Goemans and Williamson. We also obtain an improved approximation algorithm for MAX NAEf3gSAT. Finally, we provide some evidence that outward rotations can also be used to obtain improved approximation algorithms for MAX NAESAT and MAX SAT. 1 Introduction MAX CUT is perhaps the simplest and most natural APXcomplete constraint satisfaction problem (see, e.g., [AL97]). There are various simple ways of obtaining a performance guarantee of 1/2 for the problem. One of them, for example, is just choosing a random cut. No performance guarantee better than 1/2 was known for the problem until Goemans and Williamson [GW95], in a major breakthrough, used semidefinite programming to obtain an approximation algorithm ...
Mining Newsgroups Using Networks Arising From Social Behavior
, 2003
"... Recent advances in information retrieval over hyperlinked corpora have convincingly demonstrated that links carry less noisy information than text. We investigate the feasibility of applying linkbased methods in new applications domains. The specific application we consider is to partition authors ..."
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Cited by 56 (0 self)
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Recent advances in information retrieval over hyperlinked corpora have convincingly demonstrated that links carry less noisy information than text. We investigate the feasibility of applying linkbased methods in new applications domains. The specific application we consider is to partition authors into opposite camps within a given topic in the context of newsgroups. A typical newsgroup posting consists of one or more quoted lines from another posting followed by the opinion of the author. This social behavior gives rise to a network in which the vertices are individuals and the links represent "respondedto" relationships. An interesting characteristic of many newsgroups is that people more frequently respond to a message when they disagree than when they agree. This behavior is in sharp contrast to the WWW link graph, where linkage is an indicator of agreement or common interest. By analyzing the graph structure of the responses, we are able to effectively classify people into opposite camps. In contrast, methods based on statistical analysis of text yield low accuracy on such datasets because the vocabulary used by the two sides tends to be largely identical, and many newsgroup postings consist of relatively few words of text.
On the optimality of the random hyperplane rounding technique for MAX CUT
 Algorithms
, 2000
"... MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NPhard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the grap ..."
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Cited by 43 (3 self)
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MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NPhard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n dimensional sphere, and then uses a random hyperplane to cut the sphere in two, giving a cut of the graph. They show that the expected number of edges in the random cut is at least ff \Delta sdp, where ff ' 0:87856 and sdp is the value of the semidefinite program, which is an upper bound on opt, the number of edges in the maximum cut. This manuscript shows the following results: 1. The integrality ratio of the semidefinite program is ff. The previously known bound on the integrality ratio was roughly 0:8845. 2. In the presence of the so called "triangle constraints", the integrality ratio is no better than roughly 0:891. The previously known bound was above ...
Cuts for mixed 01 conic programming
, 2005
"... In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of ti ..."
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Cited by 27 (0 self)
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In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 01 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 01 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.
Bipartite subgraphs and the smallest eigenvalue
 Combinatorics, Probability & Computing
, 2000
"... Two results dealing with the relation between the smallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the smallest eigenvalue µ of any nonbipartite graph on n vertices with diameter D and maximum degree ∆ satisfies µ � − ∆ + 1 (D+1)n. This improves prev ..."
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Cited by 25 (1 self)
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Two results dealing with the relation between the smallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the smallest eigenvalue µ of any nonbipartite graph on n vertices with diameter D and maximum degree ∆ satisfies µ � − ∆ + 1 (D+1)n. This improves previous estimates and is tight up to a constant factor. The second result is the determination of the precise approximation guarantee of the max cut algorithm of Goemans and Williamson for graphs G =(V,E) in which the size of the max cut is at least AE, for all A between 0.845 and 1. This extends a result of Karloff. 1.
Improved approximation of MaxCut on graphs of bounded degree
 Electronic Colloquium on Computational Complexity ECCC 021
, 2000
"... We analyze the addition of a simple local improvement step to various known randomized approximation algorithms. Let α ∼ 0.87856 denote the best approximation ratio currently known for the Max Cut problem on general graphs [GW95]. We consider a semidefinite relaxation of the Max Cut proble ..."
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Cited by 19 (4 self)
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We analyze the addition of a simple local improvement step to various known randomized approximation algorithms. Let α ∼ 0.87856 denote the best approximation ratio currently known for the Max Cut problem on general graphs [GW95]. We consider a semidefinite relaxation of the Max Cut problem, round it using the random hyperplane rounding technique of ([GW95]), and then add a local improvement step. We show that for graphs of degree at most Δ, our algorithm achieves an approximation ratio of at least α+ε, where ε>0 is a constant that depends only on Δ. In particular, using computer assisted analysis, we show that for graphs of maximal degree 3, our algorithm obtains an approximation ratio of at least 0.921, and for 3regular graphs, the approximation ratio is at least 0.924. We note that for the semidefinite relaxation of Max Cut used in [GW95], the integrality gap is at least 1/0.884, even for 2regular graphs.