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11
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 207 (17 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
Solving Quadratic (0,1)Problems by Semidefinite Programs and Cutting Planes
, 1996
"... We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moder ..."
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Cited by 51 (7 self)
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We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moderately sized problems, having say, less than 100 binary variables, in a routine manner.
Applications of Cut Polyhedra
, 1992
"... We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole probl ..."
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Cited by 25 (2 self)
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We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of manyfermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.
On skeletons, diameters and volumes of metric polyhedra
 Combinatorics and Computer Science, Lecture
"... Abstract. We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the maxcut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency a ..."
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Cited by 15 (10 self)
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Abstract. We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the maxcut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incidence relm:ons and connectivity of the metric polytope and its relatives. In partic~dar, using its large symmetry group, we completely describe all the 13 o:bits which form the 275 840 vertices of the 21dimensional metric polytope on 7 nodes and their incidence and adjacency relations. The edge connectivity, the/skeletons and a lifting procedure valid for a large class of vertices of the metric polytope are also given. Finally, we present an ordering of the facets of a polytope, based on their adjacency relations, for the enumeration of its vertices by the double description method. 1
On the Skeleton of the Metric Polytope
, 2001
"... We consider convex polyhedra with applications to wellknown combinatorial optimization problems: the metric polytope mn and its relatives. For n # 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7  the largest previously known instan ..."
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Cited by 10 (1 self)
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We consider convex polyhedra with applications to wellknown combinatorial optimization problems: the metric polytope mn and its relatives. For n # 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7  the largest previously known instance  has 275 840 vertices but only 13 orbits. Using its large symmetry group, we enumerate orbitwise 1 550 825 600 vertices of the 28dimensional metric polytope m8 . The description consists of 533 orbits and is conjectured to be complete. The orbitwise incidence and adjacency relations are also given. The skeleton of m8 could be large enough to reveal some general features of the metric polytope on n nodes. While the extreme connectivity of the cuts appears to be one of the main features of the skeleton of mn , we conjecture that the cut vertices do not form a cutset. The combinatorial and computational applications of this conjecture are studied. In particular, a heuristic skipping the highest degeneracy is presented. 1
Node and Edge Relaxations of the MaxCut Problem
, 1994
"... We study an upper bound on the maxcut problem defined via a relaxation of the discrete problem to a continuous nonlinear convex problem, which can be solved efficiently. We demonstrate how far the approach can be pushed using advanced techniques from numerical linear algebra and nonsmooth optimizat ..."
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Cited by 6 (2 self)
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We study an upper bound on the maxcut problem defined via a relaxation of the discrete problem to a continuous nonlinear convex problem, which can be solved efficiently. We demonstrate how far the approach can be pushed using advanced techniques from numerical linear algebra and nonsmooth optimization. Various classes of graphs with up to 50,000 nodes and up to four million edges are considered. Since the theoretical bound can be computed only with a certain precision in practice, we use duality between node and edgeoriented relaxations to estimate the difference between the theoretical and the computed bounds.
On the Face Lattice of the Metric Polytope
"... In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, tailormade algorithms using their rich combinatorial features can exhib ..."
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Cited by 4 (1 self)
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In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, tailormade algorithms using their rich combinatorial features can exhibit strong performances. The main engine of these combinatorial algorithms is the use of the large symmetry group of combinatorial polytopes. Specifically we consider a polytope with applications to the wellknown maxcut and multicommodity flow problems: the metric polytope mn on n nodes. We prove that for n 9 the faces of codimension 3 of the metric polytope are partitioned into 15 orbits of its symmetry group. For n 8, we describe additional upper layers of the face lattice of mn . In particular, using the list of orbits of high dimensional faces of m8 , we prove that the description of m8 given in [9] is complete with 1 550 825 000 vertices and that the LaurentPoljak conjecture [14] holds for n 8. Many vertices of m9 are computed and additional results on the structure of the metric polytope are presented...
The Combinatorial Structure of Small Cut and Metric Polytopes
 COMBINATORICS AND GRAPH THEORY, WORLD SCIENTIFIC
, 1995
"... We study the combinatorial structure of the cut and metric polytopes on n nodes for n <= 5. Those two polytopes have a complicated geometrical structure, but using their large symmetry group, we can completely describe their face lattices. We present, for any n, some orbits of faces and give new r ..."
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Cited by 4 (4 self)
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We study the combinatorial structure of the cut and metric polytopes on n nodes for n <= 5. Those two polytopes have a complicated geometrical structure, but using their large symmetry group, we can completely describe their face lattices. We present, for any n, some orbits of faces and give new result on the tightness of the wrapping of the cut polytope by the metric polytope, disproving a conjecture of [14] on their lattices.
A counterexample to the dominating set conjecture
 Optimization Letters
"... The metric polytope metn is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities xij − xik − xjk ≤ 0 and xij + xik + xjk ≤ 2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of t ..."
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Cited by 2 (0 self)
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The metric polytope metn is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities xij − xik − xjk ≤ 0 and xij + xik + xjk ≤ 2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, in particular, for the 1 550 825 600 vertices of met8. While the overwhelming majority of the known vertices of met9 satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex. 1
OneThirdIntegrality in the MaxCut Problem
"... . Given a graph G = (V; E), the metric polytope S(G) is defined by the inequalities x(F ) \Gamma x(C n F ) jF j \Gamma 1 for F ` C; jF j odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the maxcut problem. The graph G is called 1 d integral if al ..."
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. Given a graph G = (V; E), the metric polytope S(G) is defined by the inequalities x(F ) \Gamma x(C n F ) jF j \Gamma 1 for F ` C; jF j odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the maxcut problem. The graph G is called 1 d integral if all the vertices of S(G) have their coordinates in f i d j 0 i dg. We prove that the class of 1 d integral graphs is closed under minors, and we present several minimal forbidden minors for 1 3 integrality. In particular, we characterize the 1 3 integral graphs on 7 nodes. We study several operations preserving 1 d integrality, in particular, the ksum operation for 0 k 3. We prove that series parallel graphs are characterized by the following stronger property. All vertices of the polytope S(G) " fx j ` x ug are 1 3 integral for every choice of 1 3 integral bounds `, u on the edges of G. 1991 Mathematics Subject Classification: 52B12, 90C27. Key words & Phrases: maxcut...