Sparsity of integer formulations for binary programs
Introduction
Given a non-empty set , we call a linear system , with and , an integer formulation of if . 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 , where and , is called -sparse if at most entries of are non-zero. Thus, the sparser an inequality the smaller . Using the notation , we define the sparsity function for a vector by . By overloading notation, we define for a matrix the sparsity , and finally, the minimum sparsity of a set is In particular, the minimum sparsity of is , if admits an integer formulation by -sparse inequalities but not by -sparse inequalities.
The main result of this article is an exact combinatorial characterization of the minimum sparsity of integer formulations of .
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, denotes a non-empty set. Moreover, we use the notation to denote the complement of within .
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 , let and let be the neighbors of in the -cube, i.e., where denotes the -norm. That is, the neighbors of are those -points that differ from in exactly one coordinate.
Lemma 1 Let . Then .
Proof Define and assume that , 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 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 of is called infeasible w.r.t. if no integer point in is contained in and is called maximally infeasible if it is infeasible and there does not exist an infeasible face of with . A
Separation of dense formulations
In the previous section, we have seen that sparse formulations are -hard to separate in general. By dropping the sparsity requirement, it is possible to find tractable, i.e., polynomial time separable, formulations with -coefficients on the left-hand side of many -problems. To see this, we make use of the concept of infeasibility cuts: Given a set and a point , the infeasibility or no-good cut w.r.t. is given by Observe that this
Sparsification by additional variables
Theorem 8 shows that general sets may not admit sparse integer formulations due to the structure of . If we allow to introduce additional variables, i.e., we lift the formulation of to an extended space, we will see below that always admits a -sparse formulation. In fact, this result does not exclusively hold for binary sets but for arbitrary polyhedrally representable sets. A mixed-integer set is polyhedrally representable if there exists a polyhedron with
Conclusion and outlook
We exactly characterized the maximum sparsity of inequalities that allows to define an integer formulation of and we proved that sparsest formulations are -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 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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