New Classes Of Valid Inequalities Research Articles

The Generalized Minimum Branch Vertices Problem: Properties and Polyhedral Analysis

This article introduces the Generalized Minimum Branch Vertices problem. Given an undirected graph, where the set of vertices is partitioned into clusters, the Generalized Minimum Branch Vertices problem consists of finding a tree spanning exactly one vertex for each cluster and having the minimum number of branch vertices, namely vertices with degree greater than two. When each cluster is a singleton, the problem reduces to the well-known Minimum Branch Vertices problem, which is NP-hard. We show some properties that any feasible solution to the problem has to satisfy. Some of these properties can be used to determine useless vertices or edges, which can be removed to reduce the size of the instances. We propose an integer linear programming formulation for the problem, we derive the dimension of the polytope, we study the trivial inequalities and introduce two new classes of valid inequalities, that are proved to be facet-defining.

Valid Inequalities for the Maximal Matching Polytope

Given a graph G=(V,E), a subset M of E is called a matching if no two edges in M are adjacent. A matching is said to be maximal if it is not a proper subset of any other matching. The maximal matching polytope associated with graph G is the convex hull of the incidence vectors of maximal matchings in G . In this paper, we introduce new classes of valid inequalities for the maximal matching polytope.

A new model for the asymmetric vehicle routing problem with simultaneous pickup and deliveries

The asymmetric vehicle routing problem with simultaneous pickup and deliveries is considered. This paper develops four new classes of valid inequalities for the problem. We generalize the idea of a no-good cut. Together, these help us solve 45-node randomly generated problem instances more efficiently. We report results on a set of benchmark instances in literature. In this set, we are able to show an order of magnitude improvement in computational times over currently published results in literature.

New Valid Inequalities for the Optimal Communication Spanning Tree Problem

The problem of designing a spanning tree on an underlying graph to minimize the flow costs of a given set of traffic demands is considered. Several new classes of valid inequalities are developed for the problem. Tests on 10-node problem instances show that the addition of these inequalities results in integer solutions for a significant majority of the instances without requiring any branching. In the remaining cases, root gaps of less than 1% from the optimal solutions are realized. For 30-node problem instances, the inequalities substantially reduce the number of nodes explored in the branch-and-bound tree, resulting in significantly reduced computational times. Optimal solutions are reported for problems with 30 nodes, 60 edges, fully dense traffic matrices, and Euclidean distance-based flow costs. Problems with such flow costs are well-known to be a very difficult class of problems to solve. Using the inequalities substantially improves the performance of a variable-fixing heuristic. Some polyhedral issues relating to the strength of these inequalities are also discussed. The e-companion is available at https://doi.org/10.1287/ijoc.2018.0827 .

Branch-and-Cut for the Split Delivery Vehicle Routing Problem with Time Windows

The split delivery vehicle routing problem with time windows (SDVRPTW) is a notoriously hard combinatorial optimization problem. First, it is hard to find a useful compact mixed-integer programming (MIP) formulation for the SDVRPTW. Standard modeling approaches either suffer from inherent symmetries (mixed-integer programs with a vehicle index) or cannot exactly capture all aspects of feasibility. Because of the possibility to visit customers more than once, the standard mechanisms to propagate load and time along the routes fail. Second, the lack of useful formulations has rendered any direct MIP-based approach impossible. Up to now, the most effective exact algorithms for the SDVRPTW have been branch-and-price-and-cut approaches using path-based formulations. In this paper, we propose a new and tailored branch-and-cut algorithm to solve the SDVRPTW. It is based on a new, relaxed compact model, in which some integer solutions are infeasible for the SDVRPTW. We use known and introduce some new classes of valid inequalities to cut off such infeasible solutions. One new class is path-matching constraints that generalize infeasible-path constraints. However, even with the valid inequalities, some integer solutions to the new compact formulation remain to be tested for feasibility. For a given integer solution, we build a generally sparse subnetwork of the original instance. On this subnetwork, all time-window-feasible routes can be enumerated, and a path-based residual problem then solved to decide on the selection of routes, the delivery quantities, and thereby the overall feasibility. All infeasible solutions need to be cut off. For this reason, we derive some strengthened feasibility cuts exploiting the fact that solutions often decompose into clusters. Computational experiments show that the new approach is able to prove optimality for several previously unsolved instances from the literature. The e-companion is available at https://doi.org/10.1287/trsc.2018.0825 .

Network Loading Problem: Valid inequalities from 5- and higher partitions

This paper addresses the well-known Network Loading Problem, and develops several large new classes of valid inequalities based on p-partitions of the graph. The total capacity inequalities are obtained by recursively applying a simple C-G procedure to the cut inequalities, and can be efficiently computed for p-partitions with p ≤ 10. Another class of inequalities called one-two inequalities are separated by solving a sequence of LPs, heuristically modifying the LHS coefficients of the inequality until a violated inequality is found. The spanning tree inequalities are based on a simple observation that if the demand graph of a problem is connected, the solution must be at least a tree, i.e. have (n−1) or more edges. The conditions under which these inequalities are facet-defining are analyzed. The effectiveness of these inequalities in obtaining substantially tighter bounds and reduced solutions times is demonstrated via several computational experiments. More than 10-fold reduction in the solution time is obtained on several benchmark problems.

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online. The online appendix is available at https://doi.org/10.1287/opre.2017.1650 .

On the mixed set covering, packing and partitioning polytope

We study the polyhedral structure of the mixed set covering, packing and partitioning problem, which has drawn little attention in the literature but has many real-life applications. By considering the “interactions” between the different types of edges of an induced graph, we develop new classes of valid inequalities. In particular, we derive the (generalized) mixed odd hole inequalities, and identify sufficient conditions for them to be facet-defining. In the special case when the induced graph is a mixed odd hole, the inclusion of this new facet-defining inequality provide a complete polyhedral characterization of the mixed odd hole polytope. Computational experiments indicate that these new valid inequalities may be effective in reducing the computation time in solving mixed covering and packing problems.

Capacitated Network Design using Bin-Packing

In this paper, we consider the Capacitated Network Design (CND) problem. We investigate the relationship between CND and the Bin-Packing problem. This is exploited for identifying new classes of valid inequalities for the CND problem and developing a branch-and-cut algorithm to solve it efficiently.

An exact algorithm for solving the vertex separator problem

Given G = (V, E) a connected undirected graph and a positive integer β(|V|), the vertex separator problem is to find a partition of V into no-empty three classes A, B, C such that there is no edge between A and B, max{|A|, |B|} ? β(|V|) and |C| is minimum. In this paper we consider the vertex separator problem from a polyhedral point of view. We introduce new classes of valid inequalities for the associated polyhedron. Using a natural lower bound for the optimal solution, we present successful computational experiments.

N-step mingling inequalities: new facets for the mixed-integer knapsack set

The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Program 120(2):313–346, 2009) are valid inequalities for the mixed-integer knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamtürk and Günlük (Math Program 123(2):315–338, 2010) are also derived based on MIR and they incorporate upper bounds on the integer variables. It has been shown that these inequalities are facet-defining for the mixed integer knapsack set under certain conditions and generalize several well-known valid inequalities. In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded integer variables, which we call n-step mingling inequalities (for positive integer n). These inequalities incorporate upper bounds on integer variables into n-step MIR and, therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities. Furthermore, we show that for each n, the n-step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions. For n = 2, we extend the results of Atamtürk and Günlük on facet-defining properties of 2-step mingling inequalities. For n ≥ 3, we present new facets for the mixed integer knapsack set. As a special case we also derive conditions under which the n-step MIR inequalities define facets for the mixed integer knapsack set. In addition, we show that n-step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets.

Facet-inducing inequalities for chromatic scheduling polytopes based on covering cliques

Chromatic scheduling polytopes arise as solution sets of the bandwidth allocation problem in certain radio access networks supplying wireless access to voice/data communication networks to customers with individual communication demands. This bandwidth allocation problem is a special chromatic scheduling problem; both problems are N P -complete and, furthermore, there exist no polynomial-time algorithms with a fixed quality guarantee for them. As algorithms based on cutting planes are shown to be successful for many other combinatorial optimization problems, the goal is to apply such methods to the bandwidth allocation problem. For that, knowledge on the associated polytopes is required. The present paper contributes to this issue, introducing new classes of valid inequalities based on variations and extensions of the covering-clique inequalities presented in [J. Marenco, Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems, Ph.D. Thesis, Universidad de Buenos Aires, Argentina, 2005; J. Marenco, A. Wagler, Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems, Annals of Operations Research 150-1 (2007) 159–175]. We discuss conditions ensuring that these inequalities define facets of chromatic scheduling polytopes, and we show that the associated separation problems are N P -complete.

Cycle-based facets of chromatic scheduling polytopes

Chromatic scheduling polytopes arise as solution sets of the bandwidth allocation problem in certain radio access networks, which supply wireless access to voice/data communication for customers with fixed antennas and individual demands. This problem is N P -complete and, moreover, does not admit polynomial-time approximation algorithms with a fixed quality guarantee. As algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems, our goal is to apply such methods to the bandwidth allocation problem. To gain the required knowledge on the associated polytopes, the present paper contributes by considering three new classes of valid inequalities based on cycles in the interference graph. We discuss in which cases they define facets and explore the associated separation problems, showing that two of them are solvable in polynomial time.

Acyclic Orientations with Path Constraints

Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constraints, and discuss its use in the solution of the vertex coloring problem and some versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, including proofs of which constraints of the formulation are facet-defining and the introduction of new classes of valid inequalities.

The generalized minimum spanning tree problem: Polyhedral analysis and branch‐and‐cut algorithm

AbstractThis article presents a branch‐and‐cut algorithm for the Generalized Minimum Spanning Tree Problem (GMSTP). Given an undirected graph whose vertex set is partitioned into clusters, the GMSTP consists of determining a minimum‐cost tree including exactly one vertex per cluster. Applications of the GMSTP are encountered in telecommunications. An integer linear programming formulation is presented and new classes of valid inequalities are developed, several of which are proved to be facet‐defining. A branch‐and‐cut algorithm and a tabu search heuristic are developed. Extensive computational experiments show that instances involving up to 160 or 200 vertices can be solved to optimality, depending on whether edge costs are Euclidean or random. © 2004 Wiley Periodicals, Inc.

On the capacitated lot-sizing and continuous 0–1 knapsack polyhedra

We consider the single item capacitated lot-sizing problem, a well-known production planning model that often arises in practical applications, and derive new classes of valid inequalities for it. We first derive new, easily computable valid inequalities for the continuous 0–1 knapsack problem, which has been introduced recently and has been shown to provide a useful relaxation of mixed 0–1 integer programs. We then show that by studying an appropriate continuous 0–1 knapsack relaxation of the capacitated lot-sizing problem, it is possible not only to derive new classes of valid inequalities for the capacitated lot-sizing problem, but also to derive and/or strengthen several known classes.

Wheel inequalities for stable set polytopes

We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytopePG of a graphG. Each "wheel configuration" gives rise to two such inequalities. The simplest wheel configuration is an "odd" subdivisionW of a wheel, and for these we give necessary and sufficient conditions for the wheel inequality to be facet-inducing forPW. Generalizations arise by allowing subdivision paths to intersect, and by replacing the "hub" of the wheel by a clique. The separation problem for these inequalities can be solved in polynomial time.