In this paper, we study the Cardinality Constrained Multi-cycle Problem (CCMcP) and the Car- dinality Constrained Cycle and Chain Problem (CCCCP). A feasible solution allows one or more cardinality-constrained cycles to exist on the digraph. A vertex can only be involved in at most one cycle, and there may be vertices not involved in any … Read more
The k-partition problem is an NP-hard combinatorial optimisation problem with many applications. Chopra and Rao introduced two integer programming formulations of this problem, one having both node and edge variables, and the other having only edge variables. We show that, if we take the polytopes associated with the `edge-only’ formulation, and project them into a … Read more
Del Pia and Michini recently improved the upper bound of kd due to Kleinschmidt and Onn for the largest possible diameter of the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. We introduce Euler polytopes which include a family of lattice polytopes with diameter (k+1)d/2, … Read more
In this paper we show that the diameter of a d-dimensional lattice polytope in [0,k]^n is at most (k – 1/2) d. This result implies that the diameter of a d-dimensional half-integral polytope is at most 3/2 d. We also show that for half-integral polytopes the latter bound is tight for any d. CitationUniversity of … Read more
Given a p-dimensional nonnegative, integral vector α, this paper characterizes the convex hull of the set S of nonnegative, integral vectors x that is lexicographically less than or equal to α. To obtain a finite number of elements in S, the vectors x are restricted to be component-wise upper-bounded by an integral vector u. We … Read more
Dantzig-Wolfe reformulation of an integer program convexifies a subset of the constraints, which yields an extended formulation with a potentially stronger linear programming (LP) relaxation. We would like to better understand the strength of such reformulations in general. As a first step we investigate the classical edge formulation for the stable set problem. We characterize … Read more
We investigate a new class of congestion games, called Totally Unimodular Congestion Games, in which the strategies of each player are expressed as binary vectors lying in a polyhedron defined using a totally unimodular constraint matrix and an integer right-hand side. We study both the symmetric and the asymmetric variants of the game. In the … Read more
In this article we introduce a stability concept for the coercivity of multivariate polynomials $f \in \mathbb{R}[x]$. In particular, we consider perturbations of $f$ by polynomials up to the so-called degree of stable coercivity, and we analyze this stability concept in terms of the corresponding Newton polytopes at infinity. For coercive polynomials $f \in \mathbb{R}[x]$ … Read more
The vertex cover polytopes of graphs do not admit polynomial-size extended formulations. This motivates the search for polyhedral analogues to approximation algorithms and fixed-parameter tractable (FPT) algorithms. While the polyhedral approximability of vertex cover has been studied, we know of no extended formulations parameterized by the size of the vertex cover. To this end, we … Read more
The Simple Plant Location Problem is a well-known (and NP-hard) combinatorial optimisation problem, with applications in logistics. We present a new family of valid inequalities for the associated family of polyhedra, and show that it contains an exponentially large number of new facet-defining members. We also present a new procedure, called facility augmentation, which enables … Read more