Discrete Optimization最新文献

In this paper we study the rank of polytopes contained in the 0-1 cube with respect to $t$-branch split cuts and $t$-dimensional lattice cuts for a fixed positive integer $t$. These inequalities are the same as split cuts when $t=1$ and generalize split cuts when $t>1$. For polytopes contained in the $n$-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most $n$, and this bound is tight as Cornuéjols and Li gave an example with split rank $n$. All known examples with high split rank – i.e., at least $cn$ for some positive constant $c<1$ – are defined by exponentially many (as a function of $n$) linear inequalities. For any fixed integer $t>0$, we give a family of polytopes contained in ${[0,1]}^n$ for sufficiently large $n$ such that each polytope has empty integer hull, is defined by $O\left(n\right)$ inequalities, and has rank $\Omega \left(n\right)$ with respect to $t$-dimensional lattice cuts. Therefore the split rank of these polytopes is $\Omega \left(n\right)$. It was shown earlier that there exist generalized branch-and-bound proofs, with logarithmic depth, of the nonexistence of integer points in these polytopes. Therefore, our lower bound results on split rank show an exponential separation between the depth of branch-and-bound proofs and split rank.

Given a connected graph and a subset of its vertices referred to as the sources, the minimum broadcast time problem asks for the shortest time necessary for communicating a message from the sources to all other vertices in the graph. Information exchange is possible only between neighbors, and each vertex can transmit the message to at most one neighbor at a time. Since early works on complexity theory, the problem has been known to be NP-hard. Contributions from the current text to the understanding of the minimum broadcast time problem are threefold. Through considerations of the shortest distances between sources and other vertices, a new lower bound on the broadcast time is derived. Analytical expressions of this bound are given in the single source instances of several graph classes. Fast procedures for computing upper bounds are studied next, including both the construction of feasible solutions, and the improvement of existing ones. Finally, with a focus on a new stable-set interpretation of the problem, integer programming formulations are studied, and for their theoretical interest, associated facet-defining valid inequalities are given. The computational performance of the novel methodology is evaluated in numerical experiments applied to standard benchmark instances and to instances larger than those studied in preceding recent works.

The quadratic assignment problem (QAP) is an extremely challenging NP-hard combinatorial optimization program. Due to its difficulty, a research emphasis has been to identify special cases that are polynomially solvable. Included within this emphasis are instances which are linearizable; that is, which can be rewritten as a linear assignment problem having the property that the objective function value is preserved at all feasible solutions. Various known sufficient conditions for identifying linearizable instances have been explained in terms of the continuous relaxation of a weakened version of the level-1 reformulation-linearization-technique (RLT) form that does not enforce nonnegativity on a subset of the variables. Also, conditions that are both necessary and sufficient have been given in terms of decompositions of the objective coefficients. The main contribution of this paper is the identification of a relationship between polyhedral theory and linearizability that promotes a novel, yet strikingly simple, necessary and sufficient condition for identifying linearizable instances; specifically, an instance of the QAP is linearizable if and only if the continuous relaxation of the same weakened RLT form is bounded. In addition to providing a novel perspective on the QAP being linearizable, a consequence of this study is that every linearizable instance has an optimal solution to the (polynomially-sized) continuous relaxation of the level-1 RLT form that is binary. The converse, however, is not true so that the continuous relaxation can yield binary optimal solutions to instances of the QAP that are not linearizable. Another consequence follows from our defining a maximal linearly independent set of equations in the lifted RLT variable space; we answer a recent open question that the theoretically best possible linearization-based bound cannot improve upon the level-1 RLT form.

The aim of this paper is twofold. We first provide a new orientation theorem which gives a natural and simple proof of a result of Gao and Yang (2021) on matroid-reachability-based packing of mixed arborescences in mixed graphs by reducing it to the corresponding theorem of Király (2016) on directed graphs. Moreover, we extend another result of Gao and Yang (2021) by providing a new theorem on mixed hypergraphs having a packing of mixed hyperarborescences such that their number is at least $\ell$ and at most ${\ell }^{\prime }$, each vertex belongs to exactly $k$ of them, and each vertex $v$ is the root of least $f\left(v\right)$ and at most $g\left(v\right)$ of them.

A Halin graph $G$ is a plane graph consisting of a plane embedding of a tree $T$ of order at least 4 containing no vertex of degree 2, and of a cycle $C$ connecting all leaves of $T$. Let $f_h(n,G)$ be the maximum number of copies of $G$ in a Halin graph on $n$ vertices. In this paper, we give exact values of $f_h(n,G)$ when $G$ is a path on $k$ vertices for $2\le k\le 5$. Moreover, we develop a new graph transformation preserving the number of vertices, so that the resulting graph has a monotone behavior with respect to the number of short paths.

In this paper, we study the complexity of some fundamental questions regarding box-totally dual integral (box-TDI) polyhedra. First, although box-TDI polyhedra have strong integrality properties, we prove that Integer Programming over box-TDI polyhedra is NP-complete, that is, finding an integer point optimizing a linear function over a box-TDI polyhedron is hard. Second, we complement the result of Ding et al. (2008) who proved that deciding whether a given system is box-TDI is co-NP-complete: we prove that recognizing whether a polyhedron is box-TDI is co-NP-complete.

To derive these complexity results, we exhibit new classes of totally equimodular matrices – a generalization of totally unimodular matrices – by characterizing the total equimodularity of incidence matrices of graphs.

In this paper, we discuss the computational complexities of determining optimal length refutations of infeasible integer programs (IPs). We focus on three different types of refutations, namely read-once refutations, tree-like refutations, and dag-like refutations. For each refutation type, we are interested in finding the length of the shortest possible refutation of that type. In the case of this paper, the length of a refutation is equal to the number of inferences in that refutation. The refutations in this paper are also defined by the types of inferences that can be used to derive new constraints. We are interested in refutations with two inference rules. The first rule corresponds to the summation of two constraints and is called the ADD rule. The second rule is the DIV rule which divides a constraint by a positive integer. For integer programs, we study the complexity of approximating the length of the shortest refutation of each type (read-once, tree-like, and dag-like). In this paper, we show that the problem of finding the shortest read-once refutation is NPO PB-complete. Additionally, we show that the problem of finding the shortest tree-like refutation is NPO-hard for IPs. We also show that the problem of finding the shortest dag-like refutation is NPO-hard for IPs. Finally, we show that the problems of finding the shortest tree-like and dag-like refutations are in FPSPACE.

The max-cut problem is a fundamental and much-studied $\mathrm{NP}$-hard combinatorial optimisation problem, with a wide range of applications. Several authors have shown that the max-cut problem can be solved in polynomial time if the underlying graph is free of certain minors. We give a polyhedral counterpart of these results. In particular, we show that, if a family of valid inequalities for the cut polytope satisfies certain conditions, then there is an associated minor-closed family of graphs on which the max-cut problem can be solved efficiently.

In this paper, we will answer a more general version of one of the questions proposed by Bodur et al. (2017). Specifically, we show that the $k$-aggregation closure of a covering set is a polyhedron.