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24
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
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Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 48 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
Lower bounds for lovászschrijver systems and beyond follow from multiparty communication complexity
 In Proc. 32nd Int. Conf. on Automata, Languages and Programming (ICALP'05
, 2005
"... Abstract. We prove that an ω(log 3 n) lower bound for the threeparty numberontheforehead (NOF) communication complexity of the setdisjointness function implies an n ω(1) size lower bound for treelike LovászSchrijver systems that refute unsatisfiable CNFs. More generally, we prove that an n Ω(1 ..."
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Cited by 31 (6 self)
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Abstract. We prove that an ω(log 3 n) lower bound for the threeparty numberontheforehead (NOF) communication complexity of the setdisjointness function implies an n ω(1) size lower bound for treelike LovászSchrijver systems that refute unsatisfiable CNFs. More generally, we prove that an n Ω(1) lower bound for the (k + 1)party NOF communication complexity of setdisjointness implies a 2 nΩ(1) size lower bound for all treelike proof systems whose formulas are degree k polynomial inequalities. 1
Valid inequalities for mixed integer linear programs
 Mathematical Programming B
, 2006
"... Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mi ..."
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Cited by 31 (0 self)
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Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength. Key words: mixed integer linear program, liftandproject, split cut, Gomory cut, mixed integer rounding, elementary closure, polyhedra, union of polyhedra 1.
Bounds on the Chvátal Rank of Polytopes in the 0/1Cube
"... Gomory's and Chvatal's cuttingplane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvatal rank of the polyhedron. It is wellknown that the Chvatal rank can be ..."
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Cited by 28 (1 self)
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Gomory's and Chvatal's cuttingplane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvatal rank of the polyhedron. It is wellknown that the Chvatal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1polytope. We prove that the Chvatal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the ndimensional 0/1cube is at most 3n² lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n³ lg n). Moreover, we refine this result by showing that the rank of any polytope in the 0/1cube that is defined by inequalities with small coe#cients is O(n). The latter observation explains why for most cutting planes derived in polyhedral st...
Complexity of SemiAlgebraic Proofs
, 2001
"... It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The wellstudied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the LovaszSchrijver calculi ..."
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Cited by 25 (2 self)
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It is a known approach to translate propositional formulas into systems of polynomial inequalities and to consider proof systems for the latter ones. The wellstudied proof systems of this kind are the Cutting Planes proof system (CP) utilizing linear inequalities and the LovaszSchrijver calculi (LS) utilizing quadratic inequalities. We introduce generalizations LS^d of LS that operate with polynomial inequalities of degree at most d. It turns out
Subset Algebra Lift Operators for 01 Integer Programming
, 2002
"... We extend the SheraliAdams, LovaszSchrijver, BalasCeriaCornuejols and Lasserre liftandproject methods for 01 optimization by considering liftings to subset algebras. Our methods yield polynomialtime algorithms for solving a relaxation of a setcovering problem at least as strong as that given ..."
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Cited by 19 (3 self)
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We extend the SheraliAdams, LovaszSchrijver, BalasCeriaCornuejols and Lasserre liftandproject methods for 01 optimization by considering liftings to subset algebras. Our methods yield polynomialtime algorithms for solving a relaxation of a setcovering problem at least as strong as that given by the set of all valid inequalities with small coefficients, and, more generally, all valid inequalities where the righthand side is not very large relative to the positive coefficients in the lefthand side. Applied to generalizations of vertexpacking problems, our methods yield, in polynomial time, relaxations that have unbounded rank using for example the N+ operator.
Matroid matching: the power of local search
 IN STOC
, 2010
"... We consider the classical matroid matching problem. Unweighted matroid matching for linearlyrepresented matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that ..."
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We consider the classical matroid matching problem. Unweighted matroid matching for linearlyrepresented matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that natural LP relaxations that have been studied have an Ω(n) integrality gap and, moreover, Ω(n) rounds of the SheraliAdams hierarchy are necessary to bring the gap down to a constant. More generally, for any fixed k ≥ 2 and ɛ> 0, we obtain a (k/2 + ɛ)approximation for matroid matching in kuniform hypergraphs, also known as the matroid kparity problem. As a consequence, we obtain a (k/2+ɛ)approximation for the problem of finding the maximumcardinality set in the intersection of k matroids. We also give a 3/2approximation for the weighted version of a special case of matroid matching, the matchoid problem.
Several notes on the power of GomoryChvátal cuts
, 2003
"... We prove that the Cutting Plane proof system based on GomoryChvátal cuts polynomially simulates the liftandproject system with integer coecients written in unary. The restriction on coefficients can be omitted when using Krajícek's cutfree Gentzenstyle extension of both systems. We also prov ..."
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We prove that the Cutting Plane proof system based on GomoryChvátal cuts polynomially simulates the liftandproject system with integer coecients written in unary. The restriction on coefficients can be omitted when using Krajícek's cutfree Gentzenstyle extension of both systems. We also prove that Tseitin tautologies have short proofs in this extension (of any of these systems and with any coefficients).