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An update on the Hirsch conjecture,
 Jahresber. Dtsch. Math.Ver.
, 2010
"... Abstract The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample t ..."
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Abstract The Hirsch Conjecture (1957) stated that the graph of a ddimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5dimensional polytope with 48 facets that violates a certain generalization of the dstep conjecture of Klee and Walkup.
Polyhedral graph abstractions and an approach to the linear Hirsch conjecture
, 2011
"... We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the diameters of polyhedral graphs. One particular variant has a diamet ..."
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We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the diameters of polyhedral graphs. One particular variant has a diameter which satisfies the best known upper bound on the diameters of polyhedra. Another variant has superlinear asymptotic diameter, and together with some combinatorial operations, gives a concrete approach for disproving the Linear Hirsch Conjecture.
On subdeterminants and the diameter of polyhedra
 In Proceedings of the 28th annual ACM symposium on Computational geometry, SoCG ’12
, 2012
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OBSTRUCTIONS TO WEAK DECOMPOSABILITY FOR SIMPLICIAL POLYTOPES
, 2012
"... Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facetridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that a ..."
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Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facetridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that are not weakly vertexdecomposable. These polytopes are polar to certain simple transportation polytopes. In this paper, we refine their analysis to prove that these ddimensional polytopes are not even weakly O ( √ d)decomposable. As a consequence, (weak) decomposability cannot be used to prove a polynomial version of the Hirsch Conjecture.
SUPERLINEAR SUBSET PARTITION GRAPHS WITH STRONG ADJACENCY, ENDPOINTCOUNT, AND DIMENSION REDUCTION
, 2014
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Pushing the boundaries of polytopal realizability
"... Abstract Let ∆(d, n) be the maximum possible diameter of the vertexedge graph over all ddimensional polytopes defined by n inequalities. The Hirsch bound holds for particular n and is still an open question, the best bound being the quasipolynomial one due to Kalai and Kleitman in 1992. Another ..."
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Abstract Let ∆(d, n) be the maximum possible diameter of the vertexedge graph over all ddimensional polytopes defined by n inequalities. The Hirsch bound holds for particular n and is still an open question, the best bound being the quasipolynomial one due to Kalai and Kleitman in 1992. Another natural question is for how large n and d the Hirsch bound holds. Goodey showed in 1972 that ∆(4, 10) = 5 and ∆(5, 11) = 6, and more recently, Bremner and Schewe showed ∆(4, 11) = ∆(6, 12) = 6. Here we show that ∆(4, 12) = ∆(5, 12) = 7 and present strong evidence that ∆(6, 13) = 7.
OPTIMA Mathematical Optimization Society Newsletter
, 2011
"... with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful midyear meet ..."
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with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful midyear meetings, we are now heading towards the high point of 2012: the ISMP in Berlin. I hear from good sources that preparations are progressing well, and that all augurs are favourable. As you all know, several prizes will be awarded at the ISMP opening ceremony, recognizing the contributions or both younger and more senior colleagues. You undoubtedly have seen the various calls for nominations for the Dantzig, Lagrange, Fulkerson, BealeOrchardHays and Tucker prizes as well as that for the Paul Tseng lectureship. I encourage you to seriously consider nominating one or more optimization researchers for these prizes. These awards and the high scientific standards of their recipients not only recognize