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Binary Positive Semidefinite Matrices and Associated Integer Polytopes
"... We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and wellknown integer polytopes — the cut, boolean quadric, ..."
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Cited by 3 (0 self)
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We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and wellknown integer polytopes — the cut, boolean quadric
Tabu Search  Part I
, 1989
"... This paper presents the fundamental principles underlying tabu search as a strategy for combinatorial optimization problems. Tabu search has achieved impressive practical successes in applications ranging from scheduling and computer channel balancing to cluster analysis and space planning, and more ..."
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Cited by 680 (11 self)
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term memory processes for intensifying and diversifying the search. Included are illustrative data structures for implementing the tabu conditions (and associated aspiration criteria) that underlie these processes. Part I concludes with a discussion of probabilistic tabu search and a summary
Integer points in knapsack polytopes and scovering radius
, 2012
"... Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Zm of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ Rn≥0: Ax = b} contains at least s integer points. In this paper we investigate the structure ..."
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Cited by 1 (0 self)
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Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Zm of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ Rn≥0: Ax = b} contains at least s integer points. In this paper we investigate the structure
Edmonds polytopes and a hierarchy of combinatorial problems
, 2006
"... Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integ ..."
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Cited by 170 (0 self)
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are integers, by the inequality Σaj xj ≤ a with a ≥[a0]. Obviously, if integers x1,x2,...,xn satisfy all the inequalities in S, then they satisfy also all inequalities in the closure of S. Conversely, let Σcj xj ≤ c0 hold for all choices of integers x1,x2,...,xn, that satisfy all the inequalities in S. Then we
Counting integer points in parametric polytopes using Barvinok’s rational functions
 Algorithmica
, 2007
"... Abstract Many compiler optimization techniques depend on the ability to calculate the number of elements that satisfy certain conditions. If these conditions can be represented by linear constraints, then such problems are equivalent to counting the number of integer points in (possibly) parametric ..."
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Cited by 44 (9 self)
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polytopes. It is well known that the enumerator of such a set can be represented by an explicit function consisting of a set of quasipolynomials each associated with a chamber in the parameter space. Previously, interpolation was used to obtain these quasipolynomials, but this technique has several
The number of faces of simplicial convex polytopes
 Advances in Math. 35
, 1980
"... Let P be a simplicial convex dpolytope with fi = fi(P) faces of dimension i. The vector f(P) = (f., fi,..., fdel) is called the fvector of P. In 1971 McMullen [6; 7, p. 1791 conjectured that a certain condition on a vector f = (f., fi,..., fd...J of integers was necessary and sufficient for f to ..."
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Cited by 149 (2 self)
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Let P be a simplicial convex dpolytope with fi = fi(P) faces of dimension i. The vector f(P) = (f., fi,..., fdel) is called the fvector of P. In 1971 McMullen [6; 7, p. 1791 conjectured that a certain condition on a vector f = (f., fi,..., fd...J of integers was necessary and sufficient for f
Integer Feasibility of Random Polytopes
, 2014
"... We study the ChanceConstrained Integer Feasibility Problem, where the goal is to determine whether the random polytope P (A, b) = {x ∈ Rn: Aix ≤ bi, i ∈ [m]} obtained by choosing the constraint matrix A and vector b from a known distribution is integer feasible with probability at least 1 − . We c ..."
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Cited by 1 (0 self)
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We study the ChanceConstrained Integer Feasibility Problem, where the goal is to determine whether the random polytope P (A, b) = {x ∈ Rn: Aix ≤ bi, i ∈ [m]} obtained by choosing the constraint matrix A and vector b from a known distribution is integer feasible with probability at least 1 − . We
Gap Inequalities for the Cut Polytope
 EUROPEAN J. COMBIN
, 1996
"... We introduce a new class of inequalities valid for the cut polytope, which we call gap inequalities. Each gap inequality is given by a finite sequence of integers, whose "gap" is defined as the smallest discrepancy arising when decomposing the sequence into two parts as equal as possible. ..."
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Cited by 7 (0 self)
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We introduce a new class of inequalities valid for the cut polytope, which we call gap inequalities. Each gap inequality is given by a finite sequence of integers, whose "gap" is defined as the smallest discrepancy arising when decomposing the sequence into two parts as equal as possible
The arithmetic of rational polytopes
, 2000
"... We study the number of integer points (”lattice points”) in rational polytopes. We use an associated generating function in several variables, whose coefficients are the lattice point enumerators of the dilates of a polytope. We focus on applications of this theory to several problems in combinato ..."
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Cited by 2 (0 self)
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We study the number of integer points (”lattice points”) in rational polytopes. We use an associated generating function in several variables, whose coefficients are the lattice point enumerators of the dilates of a polytope. We focus on applications of this theory to several problems
ON REGULAR POLYTOPE NUMBERS
"... Abstract. Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized in two directions. The one is a horizontal generalization which is known as polygonal number theorem, and the other is a higher dimensional generali ..."
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generalization which is known as HilberWaring problem. In this paper, we shall generalize Lagrange’s sum of four squares furthermore. To each regular polytope V in an Euclidean space, we will associate a sequence of nonnegative integers which we shall call regular polytope numbers, and consider the problem
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