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109
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 292 (18 self)
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We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
PseudoBoolean Optimization
 DISCRETE APPLIED MATHEMATICS
, 2001
"... This survey examines the state of the art of a variety of problems related to pseudoBoolean optimization, i.e. to the optimization of set functions represented by closed algebraic expressions. The main parts of the survey examine general pseudoBoolean optimization, the specially important case of ..."
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Cited by 166 (5 self)
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This survey examines the state of the art of a variety of problems related to pseudoBoolean optimization, i.e. to the optimization of set functions represented by closed algebraic expressions. The main parts of the survey examine general pseudoBoolean optimization, the specially important case of quadratic pseudoBoolean optimization (to which every pseudoBoolean optimization can be reduced), several other important special classes, and approximation algorithms.
Optimal algorithms and inapproximability results for every CSP
 In Proc. 40 th ACM STOC
, 2008
"... Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the ..."
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Cited by 143 (13 self)
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Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result: If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomialtime algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut
Relations Between Average Case Complexity and Approximation Complexity (Extended Abstract)
 In Proceedings of the 34th Annual ACM Symposium on Theory of Computing
, 2002
"... We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardne ..."
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Cited by 123 (9 self)
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We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardness of approximation results for min bisection, dense ksubgraph, max bipartite clique and the 2catalog segmentation problem. No NPhardness of approximation results are currently known for these problems.
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 101 (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. Eac ..."
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Cited by 87 (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.
The Approximability of Constraint Satisfaction Problems
, 2000
"... ... oftheoptimizationtask. Here weconsiderfourpossiblegoals: MaxCSP(MinCSP)isthe classofproblemswherethegoalistondanassignment maximizingthenumberofsatised factionproblemsdependingonthenatureofthe "underlying" constraintsaswellasonthegoal constraints(minimizingthenumberofunsatisedconstrain ..."
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Cited by 81 (1 self)
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... oftheoptimizationtask. Here weconsiderfourpossiblegoals: MaxCSP(MinCSP)isthe classofproblemswherethegoalistondanassignment maximizingthenumberofsatised factionproblemsdependingonthenatureofthe "underlying" constraintsaswellasonthegoal constraints(minimizingthenumberofunsatisedconstraints). MaxOnes(MinOnes)isthe classofoptimizationproblemswherethegoalistondan assignmentsatisfyingallconstraints withmaximum(minimum)numberofvariablesset to 1. Eachclassconsistsofinnitelymany thatdescribethepossibleconstraintsthatmaybeused. problemsandaproblemwithinaclass is specified by a finite collectionofniteBooleanfunctions pletelyclassiesalloptimizationproblems derived from Booleanconstraintsatisfaction.Our Creignou [11]. Inthisworkwedeterminetightboundsonthe "approximability"(i.e.,thera in MaxOnes,MinCSPandMinOnes.Combinedwiththeresultof Creignou,thiscomtiotowithinwhicheachproblemmay be approximatedinpolynomialtime)ofeveryproblem Tightboundsontheapproximabilityofeveryproblemin MaxCSPwereobtainedby resultscaptureadiversecollectionofoptimization problemssuchasMAX3SAT,MaxCut, (in)approximabilityoftheseoptimizationproblems andyieldacompactpresentationofmost MaxClique,MinCut,NearestCodewordetc. Ourresultsunifyrecentresultsonthe knownresults. Moreover, theseresultsprovideaformalbasistomanystatementsonthe behaviorofnaturaloptimizationproblems,thathaveso faronlybeenobservedempirically.
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 64 (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.
Gadgets, Approximation, and Linear Programming
"... We present a linear programmingbased method for finding "gadgets", i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new met ..."
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Cited by 45 (13 self)
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We present a linear programmingbased method for finding "gadgets", i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computerconstructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich and Sudan [2] of how to prove the optimality of gadgets: LP duality gives such proofs. The new gadgets, when combined with recent results of Håstad [9], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 16=17 + &epsilon; and 12=13 + &epsilon; respectively is NPhard, for every &epsilon; > 0. Prior to this work, the best known inapproximability thresholds for both problems was 71/72 [2]. Without using the gadgets from this paper, the best possible hardness that would follow from [2, 9] is 18/19. We also use the gadgets to obtain an improved approximation algorithm for MAX 3SAT which guarantees an approximation ratio of .801. This improves upon the previous best bound (implicit from [8, 5]) of .7704.