Search

We study a type of network flow problem that we call the minimum-cost F-graph ow problem. This problem generalizes the typical minimum-cost network ow problem by allowing the underlying network to be a directed hypergraph rather than a directed graph. This new problem is pertinent because it can be used to model network ow problems that occur in a dynamic, stochastic environment. We formulate this problem as an integer program, and we study specifically the case where every node has at least one outgoing edge with no capacity constraint. We show that even with this restriction, the problem of finding an integral solution is NP-hard. However, we can show that all of the inequality constraints of our formulation are either facet-defining or redundant. [ABSTRACT FROM AUTHOR]

We study stochastic integer programming models for assigning delays to flights that are destined for an airport whose capacity has been impacted by poor weather or some other exogenous factor. In the existing literature, empirical evidence seemed to suggest that a proposed integer programming model had a strong formulation, but no existing theoretical results explained the observation. We apply recent results concerning the polyhedra of stochastic network flow problems to explain the strength of the existing model, and we propose a model whose size scales better with the number of flights in the problem and that preserves the strength of the existing model. Computational results are provided that demonstrate the benefits of the proposed model. Finally, we define a type of equity property that is satisfied by both models. [ABSTRACT FROM AUTHOR]

0–1 multilinear program (MP) holds a unifying theory to LAD pattern generation. This paper studies a multi-term relaxation of the objective function of the pattern generation MP for a tight polyhedral relaxation in terms of a small number of stronger 0–1 linear inequalities. Toward this goal, we analyze data in a graph to discover useful neighborhood properties among a set of objective terms around a single constraint term. In brief, they yield a set of facet-defining inequalities for the 0–1 multilinear polytope associated with the McCormick inequalities that they replace. The construction and practical utility of the new inequalities are illustrated on a small example and thoroughly demonstrated through numerical experiments with 12 public machine learning datasets. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a mixed integer set which generalizes two well-known sets: the single node fixed-charge network set and the single arc design set. Such set arises as a relaxation of feasible sets of general mixed integer problems such as lot-sizing and network design problems. We derive several families of valid inequalities that, in particular, generalize the arc residual capacity inequalities and the flow cover inequalities. For the constant capacitated case we provide an extended compact formulation and give a partial description of the convex hull in the original space which is exact under a certain condition. By lifting some basic inequalities we provide some insight on the difficulty of obtaining such a full polyhedral description for the constant capacitated case. Preliminary computational results are presented. [ABSTRACT FROM AUTHOR]

We explore the idea of obtaining valid inequalities for a 0-1 model from a finite-domain constraint programming formulation of the problem. In particular, we formulate a graph coloring problem as a system of all-different constraints. By analyzing the polyhedral structure of all-different systems, we obtain facet-defining inequalities that can be mapped to valid cuts in the classical 0-1 model of the problem. We focus on cuts corresponding to cycles and webs and show that they are stronger than known cuts for these structures. We also identify path cuts and show they do not strengthen the bound. Computational experiments for a set of benchmark instances reveal that finite-domain cycle cuts often deliver tighter bounds, in less time, than classical 0-1 cuts. [ABSTRACT FROM AUTHOR]

One-dimensional infinite group problems have been extensively studied and have yielded strong cutting planes for mixed integer programs. Although numerical and theoretical studies suggest that group cuts can be significantly improved by considering higher-dimensional groups, there are no known facets for infinite group problems whose dimension is larger than two. In this paper, we introduce an operation that we call sequential-merge. We prove that the sequential-merge operator creates a very large family of facet-defining inequalities for high-dimensional infinite group problems using facet-defining inequalities of lower-dimensional group problems. Further, they exhibit two properties that reflect the benefits of using facets of high-dimensional group problems: they have continuous variables’ coefficients that are not dominated by those of the constituent low-dimensional cuts and they can produce cutting planes that do not belong to the first split closure of MIPs. Further, we introduce a general scheme for generating valid inequalities for lower-dimensional group problems using valid inequalities of higher-dimensional group problems. We present conditions under which this construction generates facet-defining inequalities when applied to sequential-merge inequalities. We show that this procedure yields some two-step MIR inequalities of Dash and Günlük. [ABSTRACT FROM AUTHOR]

Recent advances on the polyhedral combinatorics of the Balanced Minimum Evolution Problem (BMEP) enabled the characterization of a number of facets of its convex hull (also referred to as the BMEP polytope) as well as the discovery of connections between this polytope and the permutoassociahedron. In this article, we extend these studies, by presenting new results concerning some fundamental characteristics of the BMEP polytope, new facet-defining inequalities in the case of six or more taxa, a number of valid inequalities, and a polynomial time oracle to recognize its vertices. Our aim is to broaden understanding of the polyhedral combinatorics of the BMEP with a view to developing new and more effective exact solution algorithms. [ABSTRACT FROM AUTHOR]