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A dstep approach for distinct squares in strings
, 2011
"... We present an approach to the problem of maximum number of distinct squares in a string which underlines the importance of considering as key variables both the length n and n − d where d is the size of the alphabet. We conjecture that a string of length n and containing d distinct symbols has no mo ..."
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Cited by 9 (6 self)
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We present an approach to the problem of maximum number of distinct squares in a string which underlines the importance of considering as key variables both the length n and n − d where d is the size of the alphabet. We conjecture that a string of length n and containing d distinct symbols has no more than n − d distinct squares, show the critical role played by strings satisfying n = 2d, and present some properties satisfied by strings of length bounded by a constant times the size of the alphabet.
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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Cited by 7 (1 self)
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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.
COMBINATORICS AND GEOMETRY OF TRANSPORTATION POLYTOPES: AN UPDATE
, 2013
"... A transportation polytope consists of all multidimensional arrays or tables of nonnegative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, ..."
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Cited by 6 (1 self)
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A transportation polytope consists of all multidimensional arrays or tables of nonnegative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirtyyear update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.
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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Monotone paths in planar convex subdivisions and polytopes
, 2012
"... Consider a connected subdivision of the plane into n convex faces where every vertex is incident to at most ∆ edges. Then, starting from every vertex there is a path with at least Ω(log ∆ n) edges that is monotone in some direction. This bound is the best possible. Consider now a connected subdivisi ..."
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Consider a connected subdivision of the plane into n convex faces where every vertex is incident to at most ∆ edges. Then, starting from every vertex there is a path with at least Ω(log ∆ n) edges that is monotone in some direction. This bound is the best possible. Consider now a connected subdivision of the plane into n convex faces where exactly k faces are unbounded. Then, there is a path with at least Ω(log(n/k) / log log(n/k)) edges that is monotone in some direction. This bound is also the best possible. Our methods are constructive and lead to efficient algorithms for computing monotone paths of lengths specified above. In 3space, we show that for every n ≥ 4, there exists a polytope P with n vertices, bounded vertex degrees, and triangular faces such that every monotone path on the 1skeleton of P has at most O(log 2 n) edges. We also construct a polytope Q with n vertices, and triangular faces, (with unbounded degree however), such that every monotone path on the 1skeleton of Q has at most O(log n) edges.
COMPUTATIONAL AND STRUCTURAL APPROACHES TO PERIODICITIES IN STRINGS
, 2012
"... We investigate the function ρd(n) = max { r(x)  x is a (d, n)string} where r(x) is the number of runs in the string x, and a (d, n)string is a string with length n and exactly d distinct symbols. Our investigation is motivated by the conjecture that ρd(n) ≤ n − d. We present and discuss fundam ..."
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Cited by 2 (0 self)
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We investigate the function ρd(n) = max { r(x)  x is a (d, n)string} where r(x) is the number of runs in the string x, and a (d, n)string is a string with length n and exactly d distinct symbols. Our investigation is motivated by the conjecture that ρd(n) ≤ n − d. We present and discuss fundamental properties of the ρd(n) function. The values of ρd(n) are presented in the (d, n−d)table with rows indexed by d and columns indexed by n − d which reveals the regularities of the function. We introduce the concepts of the rcover and core vector of a string, yielding a novel computational framework for determining ρd(n) values. The computation of the previously intractable instances is achieved via first computing a lower bound, and then using the structural properties to limit our exhaustive search only to strings that can possibly exceed this number of runs. Using this approach, we extended the known maximum number of runs in binary string from 60 to 74. In doing so, we find the first examples of runmaximal strings containing four consecutive identical symbols. Our framework
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.
The Hirsch Conjecture for the Fractional Stable Set Polytope
, 2012
"... The edge formulation of the stable set problem is defined by twovariable constraints, one for each edge of a graph G, expressing the simple condition that two adjacent nodes cannot belong to a stable set. We study the fractional stable set polytope, i.e. the polytope defined by the linear relaxatio ..."
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The edge formulation of the stable set problem is defined by twovariable constraints, one for each edge of a graph G, expressing the simple condition that two adjacent nodes cannot belong to a stable set. We study the fractional stable set polytope, i.e. the polytope defined by the linear relaxation of the edge formulation. Even if this polytope is a weak approximation of the stable set polytope, its simple geometrical structure provides deep theoretical insight as well as interesting algorithmic opportunities. Exploiting a graphic characterization of the bases, we first redefine simplex pivots in terms of simple graphic operations, that turn a given basis into an adjacent one. These results lead us to prove that the Hirsch Conjecture is true for the fractional stable set polytope, i.e. the combinatorial diameter of this fractional polytope is at most equal to the number of edges of the given graph.
Finding short paths on polytopes by the shadow vertex algorithm
 In Automata, Languages, and Programming
, 2013
"... We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x ∈ Rn: Ax ≤ b} along the edges of P, where A ∈ Rm×n. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/δ that ..."
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We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x ∈ Rn: Ax ≤ b} along the edges of P, where A ∈ Rm×n. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/δ that is a measure for the flatness of the vertices of P. For integer matrices A ∈ Zm×n we show a connection between δ and the largest absolute value ∆ of any subdeterminant of A, yielding a bound of O(∆4mn4) for the length of the computed path. This bound is expressed in the same parameter ∆ as the recent nonconstructive bound of O(∆2n4 log(n∆)) by Bonifas et al. [1]. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(mn4), which significantly improves the previously best known constructive bound of O(m16n3 log3(mn)) by Dyer and Frieze [7]. 1