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We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing a superlinear… (More)

We derive a new upper bound on the diameter of the graph of a polyhedron P = {x ∈ R<sup>n</sup> : Ax ≤ b}, where A ∈ Z<sup>m×n</sup>. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by Δ. More precisely, we show that the diameter of P is bounded by O(Δ<sup>2</sup> n<sup>4</sup>… (More)

We consider a real-time scheduling problem that occurs in the design of software-based aircraft control. The goal is to distribute tasks τ i = (c i , p i) on a minimum number of identical machines and to compute offsets a i for the tasks such that no collision occurs. A task τ i releases a job of running time c i at each time a i + k · p i , k ∈ N 0 and a… (More)

We provide the currently fastest randomized (1+epsilon)-approximation algorithm for the closest lattice vector problem in the infinity-norm. The running time of our method depends on the dimension n and the approximation guarantee epsilon by 2<sup>(O(n))</sup> (log(1/epsilon))<sup>(O(n))</sup> which improves upon the (2+1/epsilon)<sup>(O(n))</sup> running… (More)

We study the simplex method over polyhedra satisfying certain " discrete curvature " lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al (SOCG 2012), Brunsch and Röglin (ICALP 2013), and Eisenbrand and Vempala (2014)… (More)

The parametric lattice-point counting problem is as follows: Given an integer matrix A ∈ Z m×n , compute an explicit formula parameterized by b ∈ R m that determines the number of integer points in the polyhedron {x ∈ R n : Ax b}. In the last decade, this counting problem has received considerable attention in the literature. Several variants of Barvinok's… (More)

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 facet-ridge graph of a simplicial polytope. Recently , De Loera and Klee provided the first examples of simplicial polytopes that are not weakly vertex-decomposable. These polytopes are… (More)

Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lovász' Local Lemma, we give a general randomized construction for subset partition graphs satisfying strong adjacency and end-point… (More)