Elsevier

Operations Research Letters

Volume 47, Issue 5, September 2019, Pages 348-352
Operations Research Letters

Sparsity of integer formulations for binary programs

https://doi.org/10.1016/j.orl.2019.06.001Get rights and content

Abstract

This paper considers integer formulations of binary sets X of minimum sparsity, i.e., the maximal number of non-zeros for each row of the corresponding constraint matrix is minimized. Providing a constructive mechanism for computing the minimum sparsity, we derive sparsest integer formulations of several combinatorial problems, including the traveling salesman problem. We also show that sparsest formulations are NP-hard to separate, while (under mild assumptions) there exists a dense formulation of X separable in polynomial time.

Introduction

Given a non-empty set X{0,1}n, we call a linear system Axb, with ARm×n and bRm, an integer formulation of X if {xZn:Axb}=X. In the following, we investigate the sparsity of such formulations. Sparsity is a desirable property of integer formulations, since it often allows optimization algorithms to perform faster in comparison with dense formulations, see, e.g., Suhl and Suhl [11], Yen et al. [12] or McCormick [9].

An inequality axβ, where aRn and βR, is called s-sparse if at most s entries of a are non-zero. Thus, the sparser an inequality the smaller s. Using the notation [n]{1,,n}, we define the sparsity function σ for a vector aRn by σ(a)|{ai0:i[n]}|. By overloading notation, we define for a matrix A the sparsity σ(A)max{σ(a):a is a row of A}, and finally, the minimum sparsity of a set X is σ(X)min(A,b){σ(A):Axb integer formulation of X}.In particular, the minimum sparsity of X is s, if X admits an integer formulation by s-sparse inequalities but not by (s1)-sparse inequalities.

The main result of this article is an exact combinatorial characterization of the minimum sparsity of integer formulations of X.

In the literature, sparsity was discussed, e.g., by Dey et al. [3], who studied the approximation of polyhedra by sparse cutting planes, and the same authors also investigated the impact of sparse cutting planes for sparse integer programs, see [4]. Among others, the approximation of polyhedra using a fixed number of dense inequalities and arbitrarily many sparse ones is investigated by Dey et al. [2] and Iroume [6].

Throughout this article, X{0,1}n denotes a non-empty set. Moreover, we use the notation X¯{0,1}nX to denote the complement of X within {0,1}n.

Section snippets

Lower bounds on minimum sparsity

To derive lower bounds on the number of non-zero coefficients needed in an integer formulation of a given set X, let xˆ{0,1}n and let N(xˆ) be the neighbors of xˆ in the 01-cube, i.e., N(xˆ){x{0,1}n:xxˆ1=1},where 1 denotes the 1-norm. That is, the neighbors of xˆ are those 01-points that differ from xˆ in exactly one coordinate.

Lemma 1

Let x̄X¯ . Then σ(X)|N(x̄)X| .

Proof

Define s=|N(x̄)X| and assume that s>1, since otherwise the statement is trivial. We assume for the sake of contradiction

Characterization of minimum sparsity

Lemma 1 allows to derive bounds on the sparsity of any integer formulation of X by a simple neighborhood argument. But this bound is not always tight. To be able to compute the minimum sparsity of an integer formulation, we introduce the concept of infeasible face coverings.

Definition 4

A face F of [0,1]n is called infeasible w.r.t.  X{0,1}n if no integer point in F is contained in X and is called maximally infeasible if it is infeasible and there does not exist an infeasible face F of [0,1]n with FF.

A

Separation of dense formulations

In the previous section, we have seen that sparse formulations are NP-hard to separate in general. By dropping the sparsity requirement, it is possible to find tractable, i.e., polynomial time separable, formulations with {0,±1}-coefficients on the left-hand side of many 01-problems. To see this, we make use of the concept of infeasibility cuts: Given a set X{0,1}n and a point x̄X¯, the infeasibility or no-good cut w.r.t. x̄ is given by i:x̄i=0xi+i:x̄i=1(1xi)1.Observe that this

Sparsification by additional variables

Theorem 8 shows that general sets X{0,1}n may not admit sparse integer formulations due to the structure of X¯. If we allow to introduce additional variables, i.e., we lift the formulation of X to an extended space, we will see below that X always admits a 3-sparse formulation. In fact, this result does not exclusively hold for binary sets but for arbitrary polyhedrally representable sets. A mixed-integer set XRn×Zq is polyhedrally representable if there exists a polyhedron PRn+q with X=P(Rn

Conclusion and outlook

We exactly characterized the maximum sparsity of inequalities that allows to define an integer formulation of X{0,1}n and we proved that sparsest formulations are NP-hard to separate in general. Complementing this result, we showed that the densest possible formulation via infeasibility cuts is always separable in polynomial time, provided the membership problem for X is polynomial time solvable. Moreover, it is possible to derive sparse formulations in an extended space.

Another interesting

Acknowledgments

We thank Tobias Fischer, Tristan Gally, Oliver Habeck, Imke Joormann, Andreas Paffenholz and Andreas M. Tillmann for fruitful discussions. We also thank four anonymous referees for valuable suggestions on an earlier version of this paper. The last two authors were partly supported by the German Research Foundation within the CRC 666.

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