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LiftandProject Ranks and Antiblocker Duality
, 2004
"... We present a very short proof of the beautiful result of Aguilera et al. that the BCCrank of the clique polytope is invariant under complementation. Such properties do not extend to the N0 and N procedures of Lovász and Schrijver, or to the N+ procedure unless P = NP. ..."
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We present a very short proof of the beautiful result of Aguilera et al. that the BCCrank of the clique polytope is invariant under complementation. Such properties do not extend to the N0 and N procedures of Lovász and Schrijver, or to the N+ procedure unless P = NP.
NOTES ON COMBINATORIAL MATHEMATICS: ANTIBLOCKING POLYHEDRA
, 1970
"... This researrh is supporlrd l»y the I'niled Stairs Air Force under Project RAND—Con ..."
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This researrh is supporlrd l»y the I'niled Stairs Air Force under Project RAND—Con
The extreme points of QSTAB(G) and its implications
, 2006
"... Perfect graphs constitute a wellstudied graph class with a rich structure, reflected by many characterizations w.r.t different concepts. Perfect graphs are, e.g., characterized as precisely those graphs G where the stable set polytope STAB(G) coincides with the clique constraint stable set polytope ..."
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polytope QSTAB(G). For all imperfect graphs STAB(G) ⊂ QSTAB(G) holds and, therefore, it is natural to measure imperfection in terms of the difference between STAB(G) and QSTAB(G). Several concepts have been developed in this direction, for instance the dilation ratio of STAB(G) and QSTAB(G) which
Combinatorial Optimization: Packing and Covering
, 2000
"... The integer programming models known as set packing and set covering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the special structure of the constraint matrix, the natural linear programming relaxation yields an optim ..."
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Cited by 51 (1 self)
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The integer programming models known as set packing and set covering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the special structure of the constraint matrix, the natural linear programming relaxation yields an optimal solution that is integer, thus solving the problem. Sometimes, both the linear programming relaxation and its dual have integer optimal solutions. Under which conditions do such integrality properties hold? This question is of both theoretical and practical interest. Minmax theorems, polyhedral combinatorics and graph theory all come together in this rich area of discrete mathematics. In addition to minmax and polyhedral results, some of the deepest results in this area come in two flavors: “excluded minor” results and “decomposition ” results. In these notes, we present several of these beautiful results. Three chapters cover minmax and polyhedral results. The next four cover excluded minor results. In the last three, we
The Sandwich Theorem
 ELECTRONIC J. COMBINATORICS
, 1994
"... This report contains expository notes about a function #(G) that is popularly known as the Lov'asz number of a graph G. There are many ways to define #(G), and the surprising variety of different characterizations indicates in itself that #(G) should be interesting. But the most interesting ..."
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Cited by 49 (0 self)
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This report contains expository notes about a function #(G) that is popularly known as the Lov'asz number of a graph G. There are many ways to define #(G), and the surprising variety of different characterizations indicates in itself that #(G) should be interesting. But the most interesting property of #(G) is probably the fact that it can be computed efficiently, although it lies "sandwiched" between other classic graph numbers whose computation is NPhard. I have tried to make these notes selfcontained so that they might serve as an elementary introduction to the growing literature on Lov'asz's fascinating function.
Comparing imperfection ratio and imperfection index for graph classes
, 2005
"... Perfect graphs constitute a wellstudied graph class with a rich structure, reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs G where the stable set polytope STAB(G) coincides with the fractional stable set polytope QSTAB ..."
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(G), the disjunctive index of QSTAB(G), and the dilation ratio of the two polytopes. Including only certain types of facets for STAB(G), we obtain graphs that are in some sense close to perfect graphs, for example minimally imperfect graphs, and certain other classes of socalled rankperfect graphs. The imperfection
On box totally dual integral polyhedra
 Mathematical Programming
, 1986
"... Edmonds and Giles introduced the class of box totally dual integral polyhedra as a generalization of submodular flow polyhedra. In this paper a geometric characterization of these polyhedra is given. This geometric result is used to show that each TDI defining system for a box TDI polyhedron is in f ..."
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is in fact a box TDI system, that the class of box TDI polyhedra is in coNP and is closed under taking projections and dominants, that the class of box perfect graphs is in coNP, and a result of Edmonds and Giles which is related to the facets of box TDI polyhdera.
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