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21
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.
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.
SEPARATING A SUPERCLASS OF COMB INEQUALITIES IN PLANAR GRAPHS
, 2000
"... Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedu ..."
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Cited by 23 (6 self)
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Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x∗ , nds one or more inequalities in the class which are violated by x , or proves that none exist. Such algorithms are at the core of the highly successful branchandcut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We de ne a generalization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges with x ∗ e ¿0 is planar. The separation algorithm runs in O(n³) time, where n is the number of vertices in the graph.
{0, 1/2}ChvatalGomory Cuts
, 1995
"... Given the integer polyhedron P I := convfx 2 Z n : Ax bg, where A 2 Z m\Thetan and b 2 Z m , a Chv'atalGomory (CG) cut is a valid inequality for P I of the type ..."
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Cited by 12 (3 self)
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Given the integer polyhedron P I := convfx 2 Z n : Ax bg, where A 2 Z m\Thetan and b 2 Z m , a Chv'atalGomory (CG) cut is a valid inequality for P I of the type
A ColumnGeneration Based BranchAndBound Algorithm For Sorting By Reversals
 MATHEMATICAL SUPPORT FOR MOLECULAR BIOLOGY; DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE 47
, 1995
"... We consider the problem of sorting a permutation by reversals (SBR), calling for the minimum number of reversals transforming a given permutation of {1, ..., n} into the identity permutation. SBR was inspired by computational biology applications, in particular genome rearrangement. We propose an ..."
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Cited by 9 (8 self)
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We consider the problem of sorting a permutation by reversals (SBR), calling for the minimum number of reversals transforming a given permutation of {1, ..., n} into the identity permutation. SBR was inspired by computational biology applications, in particular genome rearrangement. We propose an exact branchandbound algorithm for SBR. A lower bound is computed by solving a linear program with a possibly exponential (in n) number of variables, by using column generation techniques. An effective branching scheme is described, which is combined with a greedy algorithm capable of producing nearoptimal solutions. The algorithm presented can solve to optimality SBR instances of considerably larger size with respect to previous existing methods.
Wheel Inequalities for Stable Set Polytopes
, 1996
"... We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytope PG of a graph G. Each "wheel configuration" gives rise to two such inequalities. The simplest wheel configuration is an "odd" subdivision W of a wheel, and for these we give necessary and sufficie ..."
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Cited by 7 (0 self)
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We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytope PG of a graph G. Each "wheel configuration" gives rise to two such inequalities. The simplest wheel configuration is an "odd" subdivision W of a wheel, and for these we give necessary and sufficient conditions for the wheel inequality to be facetinducing for PW . Generalizations arise by allowing subdivision paths to intersect, and by replacing the "hub" of the wheel by a clique. The separation problem for these inequalities can be solved in polynomial time. 1 Introduction Let G = (V; E) be a simple connected graph with jV j = n 2 and jEj = m. A subset of V is called a stable set if it does not contain adjacent vertices of G. Let N be a stable set. The incidence vector of N is x 2 f0; 1g V such that x v = 1 if and only if v 2 N . The stable set polytope of G, denoted by PG , is the convex hull of incidence vectors of stable sets of G. Some wellknown valid inequalities for PG ...
A short proof of Guenin's characterization of weakly bipartite graphs
 J. Combin. Theory Ser. B
, 2001
"... We give a proof of Guenin's theorem characterizing weakly bipartite graphs by not having an oddK 5 minor. The proof curtails the technical and casechecking parts of Guenin's original proof. ..."
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Cited by 5 (0 self)
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We give a proof of Guenin's theorem characterizing weakly bipartite graphs by not having an oddK 5 minor. The proof curtails the technical and casechecking parts of Guenin's original proof.
On Greedy Construction Heuristics for the MaxCut Problem
 INTERNATIONAL JOURNAL ON COMPUTATONAL SCIENCE AND ENGINEERING
, 2007
"... Given a graph with nonnegative edge weights, the MAXCUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in di#erent subsets is maximized. This classical NPhard problem finds applications in VLSI design, statistical physics, an ..."
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Cited by 5 (0 self)
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Given a graph with nonnegative edge weights, the MAXCUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in di#erent subsets is maximized. This classical NPhard problem finds applications in VLSI design, statistical physics, and classification among other fields. This paper compares the performance of several greedy construction heuristics for MAXCUT problem. In particular, a new "worstout" approach is studied and the proposed edge contraction heuristic is shown to have an approximation ratio of at least 1/3. The results of experimental comparison of the worstout approach, the wellknown bestin algorithm, and modifications for both are also included.
Multicommodity flows and polyhedra
 CWI QUARTERLY
, 1993
"... Seymour's conjecture on binary clutters with the socalled weak (or Q+) maxflow mincut property implies  if true  a wide variety of results in combinatorial optimization about objects ranging from matchings to (multicommodity) flows and disjoint paths. In this paper we review in particular th ..."
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Cited by 4 (0 self)
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Seymour's conjecture on binary clutters with the socalled weak (or Q+) maxflow mincut property implies  if true  a wide variety of results in combinatorial optimization about objects ranging from matchings to (multicommodity) flows and disjoint paths. In this paper we review in particular the relation between classes of multicommodity flow problems for which the socalled cutcondition is sufficient and classes of polyhedra for which Seymour's conjecture is true.