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168
Cones of matrices and setfunctions and 01 optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1991
"... It has been recognized recently that to represent a polyhedron as the projection of a higher dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. We develop a general method to construct higherdimensional polyhedra (or, in some cases, convex sets) whose projection a ..."
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Cited by 268 (7 self)
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It has been recognized recently that to represent a polyhedron as the projection of a higher dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. We develop a general method to construct higherdimensional polyhedra (or, in some cases, convex sets) whose projection approximates the convex hull of 01 valued solutions of a system of linear inequalities. An important feature of these approximations is that one can optimize any linear objective function over them in polynomial time. In the special case of the vertex packing polytope, we obtain a sequence of systems of inequalities, such that already the first system includes clique, odd hole, odd antihole, wheel, and orthogonality constraints. In particular, for perfect (and many other) graphs, this first system gives the vertex packing polytope. For various classes of graphs, including tperfect graphs, it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets. We also discuss an extension of the method, which establishes a connection with certain submodular functions and the Möbius function of a lattice.
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 238 (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.
Semidefinite Programming Relaxations for Semialgebraic Problems
, 2001
"... A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The mai ..."
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Cited by 222 (18 self)
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A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.
Using linear programming to decode binary linear codes
 IEEE TRANS. INFORM. THEORY
, 2005
"... A new method is given for performing approximate maximumlikelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor grap ..."
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Cited by 113 (11 self)
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A new method is given for performing approximate maximumlikelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor graph or paritycheck representation of the code. The resulting “LP decoder” generalizes our previous work on turbolike codes. A precise combinatorial characterization of when the LP decoder succeeds is provided, based on pseudocodewords associated with the factor graph. Our definition of a pseudocodeword unifies other such notions known for iterative algorithms, including “stopping sets, ” “irreducible closed walks, ” “trellis cycles, ” “deviation sets, ” and “graph covers.” The fractional distance ��— ™ of a code is introduced, which is a lower bound on the classical distance. It is shown that the efficient LP decoder will correct up to ��— ™ P I errors and that there are codes with ��— ™ a @ I A. An efficient algorithm to compute the fractional distance is presented. Experimental evidence shows a similar performance on lowdensity paritycheck (LDPC) codes between LP decoding and the minsum and sumproduct algorithms. Methods for tightening the LP relaxation to improve performance are also provided.
MAP estimation via agreement on (hyper)trees: Messagepassing and linear programming approaches
 IEEE Transactions on Information Theory
, 2002
"... We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of treestructured problems, we obtain an upper bound on the optimal value of the original ..."
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Cited by 107 (11 self)
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We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of treestructured problems, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is met with equality if and only if the tree problems share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original problem. Next we present and analyze two methods for attempting to obtain tight upper bounds: (a) a treereweighted messagepassing algorithm that is related to but distinct from the maxproduct (minsum) algorithm; and (b) a treerelaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds. Finally, we discuss the conditions that govern when the relaxation is tight, in which case the MAP configuration can be obtained. The analysis described here generalizes naturally to convex combinations of hypertreestructured distributions.
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 99 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Twoprover oneround proof systems: their power and their problems
 In Proceedings of the TwentyFourth Annual ACM Symposium on Theory of Computing
, 1992
"... We characterize the power of twoprover oneround (MI’P(2, 1)) proof systems, showing that M1P(2, 1) = NEXPTIME. However, the following intriguing question remains open: Does parallel repetition decrease the error probability y of MlP(2, 1) proof systems? We use techniques based on quadratic program ..."
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Cited by 96 (5 self)
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We characterize the power of twoprover oneround (MI’P(2, 1)) proof systems, showing that M1P(2, 1) = NEXPTIME. However, the following intriguing question remains open: Does parallel repetition decrease the error probability y of MlP(2, 1) proof systems? We use techniques based on quadratic programming to study this problem, and prove the parallel repetition conjecture in some special cases. Interestingly, our work leads to a general polynomial time heuristic for any NPproblem. We prove the effectiveness of this heuristic for several problems, such as computing the chromatic number of perfect graphs. him, without the ability to communicate with other provers. When the protocol ends, the verifier evaluates a polynomial time predicate on his coin tosses and on the messages exchanged and decides whether to accept or reject. Definition 1.1 A kprover rround protocol is an MIP(k,r) proof system \or language L ij;
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
"... ..."
Decoding ErrorCorrecting Codes via Linear Programming
, 2003
"... Abstract. Errorcorrecting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original informa ..."
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Cited by 79 (6 self)
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Abstract. Errorcorrecting codes are fundamental tools used to transmit digital information over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original information up to the full errorcorrecting potential of the system is often very complex, especially for modern codes that approach the theoretical limits of the communication channel. In this thesis we investigate the application of linear programming (LP) relaxation to the problem of decoding an errorcorrecting code. Linear programming relaxation is a standard technique in approximation algorithms and operations research, and is central to the study of efficient algorithms to find good (albeit suboptimal) solutions to very difficult optimization problems. Our new “LP decoders ” have tight combinatorial characterizations of decoding success that can be used to analyze errorcorrecting performance. Furthermore, LP decoders have the desirable (and rare) property that whenever they output a result, it is guaranteed to be the optimal result: the most likely (ML) information sent over the
Proving Integrality Gaps Without Knowing the Linear Program
 Theory of Computing
, 2002
"... Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods see ..."
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Cited by 56 (2 self)
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Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods seem relevant to other problems as well.