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Treewidth and the SheraliAdams operator
, 2003
"... We describe a connection between the treewidth of graphs and the SheraliAdams reformulation procedure for 0/1 integer programs. For the case of vertex packing problems, our main result can be restated as follows: let G be a graph, let k 1 and let x be a feasible vector for the formulation ..."
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We describe a connection between the treewidth of graphs and the SheraliAdams reformulation procedure for 0/1 integer programs. For the case of vertex packing problems, our main result can be restated as follows: let G be a graph, let k 1 and let x be a feasible vector for the formulation produced by applying the levelk SheraliAdams algorithm to the edge formulation for STAB(G). Then for any subgraph H of G, of treewidth at most k, the restriction of x to R V (H) is a convex combination of stable sets of H. 1
On Disjunctive Cuts for Combinatorial Optimization
 J. OF COMB. OPT
, 2000
"... In the successful branchandcut approach to combinatorial optimization, linear inequalities are used as cutting planes within a branchandbound framework. Although researchers often prefer to use facetinducing inequalities as cutting planes, good computational results have recently been obtaine ..."
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Cited by 4 (1 self)
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In the successful branchandcut approach to combinatorial optimization, linear inequalities are used as cutting planes within a branchandbound framework. Although researchers often prefer to use facetinducing inequalities as cutting planes, good computational results have recently been obtained using disjunctive cuts, which are not guaranteed to be facetinducing in general. A partial explanation for the success of the disjunctive cuts is given in this paper. It is shown that, for six important combinatorial optimization problems (the clique partitioning, maxcut, acyclic subdigraph, linear ordering, asymmetric travelling salesman and set covering problems), certain facetinducing inequalities can be obtained by simple disjunctive techniques. New polynomialtime separation algorithms are obtained for these inequalities as a byproduct. The disjunctive approach is then compared and contrasted with some other `generalpurpose' frameworks for generating cutting planes and some conclusions are made with respect to the potential and limitations of the disjunctive approach.
Weak Order Polytopes
, 1999
"... imary: 52B12, 52B15; secondary: 90A06, 90A08. OR/MS Index 1978 subject classification. Primary: Decision Theory, Programming. Key words. Weak order, partition, polytope, preference, choice probability 1 1. Introduction. The present study explicates structural aspects of the weak order polytope P ..."
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Cited by 2 (2 self)
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imary: 52B12, 52B15; secondary: 90A06, 90A08. OR/MS Index 1978 subject classification. Primary: Decision Theory, Programming. Key words. Weak order, partition, polytope, preference, choice probability 1 1. Introduction. The present study explicates structural aspects of the weak order polytope P n WO for n 2. The vertices of P n WO represent members of the family of reflexive weak orders  on n = f1; 2; : : : ; ng. Our motivation is threefold. The first aspect is the preeminence of weak orders in theories of preference, comparative probability, and social choice as represented during the past century by de Finetti (1937), von Neumann and Morgenstern (1944), Arrow (1963), Savage (1954), Anscombe and Aumann (1963), Fishburn (1970, 1973) and Wakker (1989). The second is probabili
Transitive packing: A unifying concept in combinatorial optimization
, 2002
"... This paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately. It introduces the notion of transitive packing and the transitive packing polytope. Polytopes that turn out to be special cases of the transitive packing polytope include t ..."
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This paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately. It introduces the notion of transitive packing and the transitive packing polytope. Polytopes that turn out to be special cases of the transitive packing polytope include the node packing, acyclic subdigraph, bipartite subgraph, planar subgraph, clique partitioning, partition, transitive acyclic subdigraph, interval order, and relatively transitive subgraph polytopes. We give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope, thereby introducing generalized cycle, generalized clique, generalized antihole, generalized antiweb, and odd partition inequalities. On the one hand, these classes subsume several known classes of valid inequalities for several special cases; on the other hand, they yield many new inequalities for several other special cases. For some of the classes we also prove a lower bound on their Gomory–Chvátal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering, as well as to balanced and ideal matrices.