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A semidefinite programming based polyhedral cut and price approach for the maxcut problem
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
, 2006
"... We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the wellknown SDP relaxation of maxcut is formulated as a semiinfinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; this co ..."
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Cited by 16 (6 self)
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We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the wellknown SDP relaxation of maxcut is formulated as a semiinfinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; this constitutes the pricing (column generation) phase of the algorithm. Cutting planes based on the polyhedral theory of the maxcut problem are then added to the primal problem in order to improve the SDP relaxation; this is the cutting phase of the algorithm. We provide computational results, and compare these results with a standard SDP cutting plane scheme.
Vertex Cover resists SDPs tightened by local hypermetric inequalities
, 2007
"... We consider the standard semidefinite programming (SDP) relaxation for the vertex cover problem to which all hypermetric inequalities supported on at most k vertices are added and show that the integrality gap for such SDPs remains 2 − o(1) even for k = O ( � log n / loglog n). This extends results ..."
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Cited by 6 (3 self)
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We consider the standard semidefinite programming (SDP) relaxation for the vertex cover problem to which all hypermetric inequalities supported on at most k vertices are added and show that the integrality gap for such SDPs remains 2 − o(1) even for k = O ( � log n / loglog n). This extends results by KleinbergGoemans, Charikar and Hatami et al. who considered vertex cover SDPs tightened using the triangle and pentagonal inequalities, respectively. Our result is complementary to a recent result by Georgiou et al. proving integrality gaps for vertex cover SDPs in the LovászSchrijver hierarchy. However, the SDPs we consider are incomparable to the SDPs analyzed by Georgiou et al. In particular we show that vertex cover SDPs in the LovászSchrijver hierarchy fail to satisfy any hypermetric constraints supported on an independent set of the input graph. This constrasts with the LP LovászSchrijver hierarchy where all local LP constraints are derived.
On the Complexity of Testing Hypermetric, Negative Type, kGonal And Gap Inequalities
 DISCRETE AND COMPUTATIONAL GEOMETRY. LECTURE NOTES IN COMPUTER SCIENCE
, 2003
"... Hypermetric inequalities have many applications, most recently in the approximate solution of maxcut problems by linear and semidefinite programming. However, not much is known about the separation problem for these inequalities. Previously Avis and Grishukhin showed that certain special cases o ..."
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Cited by 3 (0 self)
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Hypermetric inequalities have many applications, most recently in the approximate solution of maxcut problems by linear and semidefinite programming. However, not much is known about the separation problem for these inequalities. Previously Avis and Grishukhin showed that certain special cases of the separation problem for hypermetric inequalities are NPhard, as evidence that the separation problem is itself hard. In this paper we show that similar results hold for inequalities of negative type, even though the separation problem for negative type inequalities is well known to be solvable in polynomial time. We also show similar results hold for the more general kgonal and gap inequalities.
Robust algorithms for Max Independent Set on Minorfree graphs based on the SheraliAdams Hierarchy
"... Abstract. This work provides a Linear Programmingbased Polynomial Time Approximation Scheme (PTAS) for two classical NPhard problems on graphs when the input graph is guaranteed to be planar, or more generally Minor Free. The algorithm applies a sufficiently large number (some function of 1/ɛ when ..."
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Abstract. This work provides a Linear Programmingbased Polynomial Time Approximation Scheme (PTAS) for two classical NPhard problems on graphs when the input graph is guaranteed to be planar, or more generally Minor Free. The algorithm applies a sufficiently large number (some function of 1/ɛ when 1 + ɛ approximation is required) of rounds of the socalled SheraliAdams LiftandProject system. needed to obtain a (1 + ɛ)approximation, where f is some function that depends only on the graph that should be avoided as a minor. The problem we discuss are the wellstudied problems, the Max Independent Set and Min Vertex Cover problems. An curious fact we expose is that in the world of minorfree graph, the Min Vertex Cover is harder in some sense than the Max Independent Set. Our main result shows how to get a PTAS for Max Independent Set in the more general “noisy setting ” in which input graphs are not assumed to be planar/minorfree, but only close to being so. In this setting we bound integrality gaps by 1+ɛ, which in turn provides a 1+ɛ approximation of the optimum value; however we don’t know how to actually find a solution with this approximation guarantee. While there are known combinatorial algorithms for the nonnoisy setting of the above graph problems, we know of no previous approximation algorithms in the noisy setting. Further, we give evidence that current combinatorial techniques will fail to generalize to this noisy setting. 1
Semidefinite relaxations for integer programming
, 2009
"... We survey some recent developments in the area of semidefinite optimization applied to integer programming. After recalling some generic modeling techniques to obtain semidefinite relaxations for NPhard problems, we look at the theoretical power of semidefinite optimization in the context of the Ma ..."
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We survey some recent developments in the area of semidefinite optimization applied to integer programming. After recalling some generic modeling techniques to obtain semidefinite relaxations for NPhard problems, we look at the theoretical power of semidefinite optimization in the context of the MaxCut and the Coloring Problem. In the second part, we consider algorithmic questions related to semidefinite optimization, and point to some recent ideas to handle large scale problems. The survey is concluded with some more advanced modeling techniques, based on matrix relaxations leading to copositive matrices.
Exploring the Relationship Between MaxCut and Stable Set Relaxations
, 2005
"... The maxcut and stable set problems are two fundamental N Phard problems in combinatorial optimization. It has been known for a long time that any instance of the stable set problem can be easily transformed into a maxcut instance. Moreover, Laurent, Poljak, Rendl and others have shown that any co ..."
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The maxcut and stable set problems are two fundamental N Phard problems in combinatorial optimization. It has been known for a long time that any instance of the stable set problem can be easily transformed into a maxcut instance. Moreover, Laurent, Poljak, Rendl and others have shown that any convex set containing the cut polytope yields, via a suitable projection, a convex set containing the stable set polytope. We review this work, and then extend it in the following ways. We show that the rounded version of certain ‘positive semidefinite ’ inequalities for the cut polytope imply, via the same projection, a surprisingly large variety of strong valid inequalities for the stable set polytope. These include the clique, odd hole, odd antihole, web and antiweb inequalities, and various inequalities obtained from these via sequential lifting. We also examine a less general class of inequalities for the cut polytope, which we call odd clique inequalities, and show that they are, in general, much less useful for generating valid inequalities for the stable set polytope. As well as being of theoretical interest, these results have algorithmic implications. In particular, we obtain as a byproduct a polynomialtime separation algorithm for a class of inequalities which includes all web inequalities.
On the Tightening of the Standard SDP for Vertex Cover with ℓ1 Inequalities
 LIPICS LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
"... We show that the integrality gap of the standard SDP for VERTEX COVER on instances of n vertices remains 2 − o(1) even after the addition of all hypermetric inequalities. Our lower bound requires new insights into the structure of SDP solutions behaving like ℓ1 metric spaces when one point is remove ..."
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We show that the integrality gap of the standard SDP for VERTEX COVER on instances of n vertices remains 2 − o(1) even after the addition of all hypermetric inequalities. Our lower bound requires new insights into the structure of SDP solutions behaving like ℓ1 metric spaces when one point is removed. We also show that the addition of all ℓ1 inequalities eliminates any solutions that are not convex combination of integral solutions. Consequently, we provide the strongest possible separation between hypermetrics and ℓ1 inequalities with respect to the tightening of the standard SDP for VERTEX COVER.