Discrete Optimization最新文献

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 maximum number of edges in a graph with matching number $m$ and maximum degree $d$ has been determined in Chvátal and Hanson (1976) and Balachandran and Khare (2009), where some extremal graphs have also been provided. Then, a new question has emerged: how the maximum edge count is affected by forbidding some subgraphs occurring in these extremal graphs? In Ahanjideh et al. (2022), the problem is solved in triangle-free graphs for $d\ge m$, and for $d<m$ with either $Z\left(d\right)\le m<2d$ or $d\le 6$, where $Z\left(d\right)$ is approximately $5d/4$. The authors derived structural properties of triangle-free extremal graphs, which allows us to focus on constructing small extremal components to form an extremal graph. Based on these findings, in this paper, we develop an integer programming formulation for constructing extremal graphs. Since our formulation is highly symmetric, we use our own implementation of Orbital Branching to reduce symmetry. We also implement our integer programming formulation so that the feasible region is restricted iteratively. Using a combination of the two approaches, we expand the solution into $d\le 10$ instead of $d\le 6$ for $m>d$. Our results endorse the formula for the number of edges in all extremal triangle-free graphs conjectured in Ahanjideh et al. (2022).

The problem of minimizing a multilinear function of binary variables is a well-studied NP-hard problem. The set of solutions of the standard linearization of this problem is called the multilinear set. We study a cardinality constrained version of it with upper and lower bounds on the number of nonzero variables. We call the set of solutions of the standard linearization of this problem a multilinear set with cardinality constraints. We characterize a set of conditions on these multilinear terms (called properness) and observe that under these conditions the convex hull description of the set is tractable via an extended formulation. We then give an explicit polyhedral description of the convex hull when the multilinear terms have a nested structure. Our description has an exponential number of inequalities which can be separated in polynomial time. Finally, we generalize these inequalities to obtain valid inequalities for the general case.

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.

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.

We revisit online weighted edge coloring. In this problem, weighted edges of a graph are presented one by one, to be colored with positive integers. It is required that for every vertex, all its edges of every common color will have a total weight not exceeding 1. We provide an improved upper bound on the performance of a greedy algorithm First Fit for the case of arbitrary weights, and for the case of weights not exceeding $\frac{1}{2}$. Here, the meaning of First Fit is that every edge is colored with a color of the smallest index that will keep the coloring valid. This improves the state-of-the-art with respect to online algorithms for this variant of edge coloring. We also show new lower bounds on the performance of any online algorithm with weights in $(0,\frac{1}{t}]$, for any integer $t\ge 2$.

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.

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.