Elsevier

Operations Research Letters

Volume 44, Issue 5, September 2016, Pages 625-629
Operations Research Letters

Geometric proofs for convex hull defining formulations

https://doi.org/10.1016/j.orl.2016.07.006Get rights and content

Abstract

A conjecture appeared recently in Cacchiani et al. (2013) that a proposed LP relaxation of a certain integer programming problem defines the convex hull of its integer points. We review a little known technique described in Zuckerberg (2004) that can be used to construct geometric proofs that an LP relaxation is convex hull defining. In line with this technique, we show that their conjecture is correct.

Introduction

The connection between probability measures and integer programming has antecedents in  [6] (in the “Remark” on page 186), and earlier (see Part 1 in  [4]), with later development in  [2] (beginning in Section 2.1) and in  [8] (Chapters 3 and 4), where this connection is shown to generalize and contextualize the lifting methods of  [1], [7], [6], [5]. In the course of the analysis in  [8], a method is described by which an LP relaxation of an integer programming formulation can be proven to define the convex hull of its integer points if it can be shown that for each point in Rn that is in the linear relaxation, it is possible to draw n sets in an arbitrary measure space that have properties that match the logical properties of the integer feasible set in a certain way. The method is described as “geometric”, or as in  [8], as the “proof by picture method”, as one can potentially demonstrate that a formulation is convex hull defining by physically drawing sets in the real line or plane with the requisite properties. In this work we will provide a compact description and rigorous justification of the method, and we will use it to resolve a conjecture that appeared in  [3].

Section snippets

Binary integer programming and probability measures

First we review some basic definitions and facts.

Definition 1

  • 1.

    Given a set U and a family L of subsets of U, the pair Q=(U,L) is said to be an “algebra on U” if L  is closed under unions, intersections and complementations, where complementation is defined as hcUh,hL. Note that for any algebra Q=(U,L),UL. Given an algebra Q=(U,L), we will refer to a set h as being “in algebra Q” if hL.

  • 2.

    A nonempty set a in an algebra Q=(U,L) is called an “atom” of Q  if for every hL, either ah=a or ah=, i.e.  a

Proving the conjecture

We now have the machinery to prove the conjecture of  [3]. The set F  for which we will be seeking a convex hull formulation is the set of 0,1 points in Rn that satisfy a covering constraint i=1naixib, and for which exactly two coordinates have value 1. The coefficients of the covering constraint are listed in descending order a1a2an0. (The convex hull of F  is referred to in  [3] as Pˆcover.)

In line with the notation of  [3], we define t{1,,n1} to be the largest index such that at+a

Concluding remarks and further work

Much of the appeal of the proof by picture method as an aid in proving the integrality of formulations is that it may be used to effectively transform the geometry of an n dimensional problem to one or two dimensions, where most people have far better intuition. It is useful in particular where the feasible space has a simple set theoretic (i.e. logical) interpretation. Though we have not discussed this here, it can therefore also be helpful in supplying geometric intuition when dealing with

Acknowledgments

I would like to express my gratitude to Dan Bienstock (Columbia University), who has collaborated with me on topics related to this work for many years, and to my colleague Peter Malkin (BHP Billiton, Melbourne University). Their insightful comments and suggestions have significantly enhanced this work, and are much appreciated. I also gratefully acknowledge the ongoing support of BHP Billiton.

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