Stochastic lot-sizing problem with deterministic demands and Wagner-Whitin costs

In this paper, we consider a two-stage stochastic uncapacitated lot-sizing problem with deterministic demands and Wagner-Whitin costs. We develop an extended formulation in the higher dimensional space that provides integral solutions by showing that its constraint matrix is totally unimodular. We also provide the integral polyhedron of the problem in the original space by projecting the extended formulation to the original space.     

Some formulations for the group steiner tree problem

The group Steiner tree problem consists of, given a graph G, a collection R of subsets of V(G) and a cost c(e) for each edge of G, finding a minimum-cost subtree that connects at least one vertex from each R∈R. It is a generalization of the well-known Steiner tree problem that arises naturally in the design of VLSI chips. In this paper, we study a polyhedron associated with this problem and some extended formulations. We give facet defining inequalities and explore the relationship between the group Steiner tree problem and other combinatorial optimization problems.     

Generalised 2-circulant inequalities for the max-cut problem

The max-cut problem is a fundamental combinatorial optimisation problem, with many applications. Poljak and Turzik found some facet-defining inequalities for the associated polytope, which we call 2-circulant inequalities. We present a more general family of facet-defining inequalities, an exact separation algorithm that runs in polynomial time, and some computational results.     

On solving large instances of the capacitated facility location problem

We present two sets of results pertaining to the solution of capacitated facility location problems that are large, especially with regard to the number of customers. One set of results relates to customer aggregation, while another set of results concerns the judicious selection of variable-upper-bounding (VUB) constraints to include in the initial integer-programming formulation.     

Polyhedral results on single node variable upper-bound flow models with allowed configurations

In this paper we investigate the convex hull of single node variable upper-bound flow models with allowed configurations. Such a model is defined by a set , where ρ is one of , = or , and Z{0,1}ⁿ consists of the allowed configurations. We consider the case when Z consists of affinely independent vectors. Under this assumption, a characterization of the non-trivial facets of the convex hull of X_ρ(Z) for each relation ρ is provided, along with polynomial time separation algorithms. Applications in scheduling and network design are also discussed.     

Chromatic scheduling polytopes coming from the bandwidth allocation problem in point-to-multipoint radio access systems

Point-to-Multipoint systems are a kind of radio systems supplying wireless access to voice/data communication networks. Such systems have to be run using a certain frequency spectrum, which typically causes capacity problems. Hence it is, on the one hand, necessary to reuse frequencies but, on the other hand, no interference must be caused thereby. This leads to a combinatorial optimization problem, the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard and it is known that, for these problems, there exist no polynomial time algorithms with a fixed approximation ratio. Algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems. In order to apply such methods, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring basic properties of chromatic scheduling polytopes and several classes of facet-defining inequalities. J. L. Marenco: This work supported by UBACYT Grant X036, CONICET Grant 644/98 and ANPCYT Grant 11-09112. A. K. Wagler: This work supported by the Deutsche Forschungsgemeinschaft (Gr 883/9–1).     

Two graph isomorphism polytopes

The convex hull ψ_n,n of certain (n!)² tensors was considered recently in connection with graph isomorphism. We consider the convex hull ψ_n of the n! diagonals among these tensors. We show: 1. The polytope ψ_n is a face of ψ_n,n. 2. Deciding if a graph G has a subgraph isomorphic to H reduces to optimization over ψ_n. 3. Optimization over ψ_n reduces to optimization over ψ_n,n. In particular, this implies that the subgraph isomorphism problem reduces to optimization over ψ_n,n.     

On the combinatorial structure of chromatic scheduling polytopes

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 for customers with individual communication demands. To maintain the links, only frequencies from a certain spectrum can be used, which typically causes capacity problems. Hence it is necessary to reuse frequencies but no interference must be caused by this reuse. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard, and there do not even exist polynomial time algorithms with a fixed quality guarantee.As algorithms based on cutting planes have 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, exploring the combinatorial structure of chromatic scheduling polytopes for increasing frequency spans. We observe that the polytopes pass through various stages—emptyness, non-emptyness but low-dimensionality, full-dimensionality but combinatorial instability, and combinatorial stability—as the frequency span increases. We discuss the thresholds for this increasing “quantity” giving rise to a new combinatorial “quality” of the polytopes, and we prove bounds on these thresholds. In particular, we prove combinatorial equivalence of chromatic scheduling polytopes for large frequency spans and we establish relations to the linear ordering polytope.     

Relating single-scenario facets to the convex hull of the extensive form of a stochastic single-node flow polytope

We study when a facet-defining inequality for a deterministic, single-scenario subproblem is also facet-defining for the extensive form of a two-stage stochastic mixed-integer linear program (SMIP). To answer this question, we introduce a novel stochastic variant of the well-known single-node flow (SNF) polytope, and present necessary and sufficient conditions for single-scenario facet-defining inequalities to be facet-defining for the extensive form. We further demonstrate that our stochastic SNF polytope is a relaxation of a broad subclass of SMIPs, illustrating its generality.     

The node-edge weighted 2-edge connected subgraph problem: Linear relaxation, facets and separation

Let G=(V,E) be a undirected k-edge connected graph with weights c_e on edges and w_v on nodes. The minimum 2-edge connected subgraph problem, 2ECSP for short, is to find a 2-edge connected subgraph of G, of minimum total weight. The 2ECSP generalizes the well-known Steiner 2-edge connected subgraph problem. In this paper we study the convex hull of the incidence vectors corresponding to feasible solutions of 2ECSP. First, a natural integer programming formulation is given and it is shown that its linear relaxation is not sufficient to describe the polytope associated with 2ECSP even when G is series-parallel. Then, we introduce two families of new valid inequalities and we give sufficient conditions for them to be facet-defining. Later, we concentrate on the separation problem. We find polynomial time algorithms to solve the separation of important subclasses of the introduced inequalities, concluding that the separation of the new inequalities, when G is series-parallel, is polynomially solvable.     

New valid inequalities for the fixed-charge and single-node flow polytopes

The most effective software packages for solving mixed 0–1linear programs use strong valid linear inequalities derived from polyhedral theory. We introduce a new procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the fixed-charge and single-node flow polytopes. The resulting inequalities are very different from the previously known inequalities (such as flow cover and flow pack inequalities), and define facets under certain conditions.     

Facets of the polytope of the asymmetric travelling salesman problem with replenishment arcs

The Asymmetric Travelling Salesman Problem with Replenishment Arcs (RATSP) is a new class of problems arising from work related to aircraft routing. Given a digraph with cost on the arcs, a solution of the RATSP, like that of the Asymmetric Travelling Salesman Problem, induces a directed tour in the graph which minimises total cost. However the tour must satisfy additional constraints: the arc set is partitioned into replenishment arcs and ordinary arcs, each node has a non-negative weight associated with it, and the tour cannot accumulate more than some weight limit before a replenishment arc must be used. To enforce this requirement, constraints are needed. We refer to these as replenishment constraints.In this paper, we review previous polyhedral results for the RATSP and related problems, then prove that two classes of constraints developed in V. Mak and N. Boland [Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs, Technical Report TR M05/03, School of Information Technology, Deakin University, 2005] are, under appropriate conditions, facet-defining for the RATS polytope.     

On packing and covering polyhedra of consecutive ones circulant clutters

Building on work by G. Cornuéjols and B. Novick and by L. Trotter, we give different characterizations of contractions of consecutive ones circulant clutters that give back consecutive ones circulant clutters. Based on a recent result by G. Argiroffo and S. Bianchi, we then arrive at characterizations of the vertices of the fractional set covering polyhedron of these clutters. We obtain similar characterizations for the fractional set packing polyhedron using a result by F.B. Shepherd, and relate our findings with similar ones obtained by A. Wagler for the clique relaxation of the stable set polytope of webs. Finally, we show how our results can be used to obtain some old and new results on the corresponding fractional set covering polyhedron using properties of Farey series. Our results do not depend on Lehman’s work or blocker/antiblocker duality, as is traditional in the field.     

Valid inequalities for mixed-integer programmes with fixed charges on sets of variables

We consider mixed 0-1 linear programmes in which one is given a collection of (not necessarily disjoint) sets of variables and, for each set, a fixed charge is incurred if and only if at least one of the variables in the set takes a positive value. We derive strong valid linear inequalities for these problems, and show that they generalise and dominate a subclass of the well-known flow cover inequalities for the classical fixed-charge problem.     

On the monotonization of polyhedra

In polyhedral combinatorics one often has to analyze the facial structure of less than full dimensional polyhedra. The presence of implicit or explicit equations in the linear system defining such a polyhedron leads to technical difficulties when analyzing its facial structure. It is therefore customary to approach the study of such a polytopeP through the study of one of its (full dimensional) relaxations (monotonizations) known as the submissive and the dominant ofP. Finding sufficient conditions for an inequality that induces a facet of the submissive or the dominant of a polyhedron to also induce a facet of the polyhedron itself has been posed in the literature as an important research problem. Our paper goes a long way towards solving this problem. We address the problem in the framework of a generalized monotonization of a polyhedronP, g-mon(P), that subsumes both the submissive and the dominant, and give a sufficient condition for an inequality that defines a facet of g-mon(P) to define a facet ofP. For the important cases of the traveling salesman (TS) polytope in both its symmetric and asymmetric variants, and of the linear ordering polytope, we give sufficient conditions trivially easy to verify, for a facet of the monotone completion to define a facet of the original polytope itself. Research supported by grant DMI-9201340 of the National Science Foundation and contract N00014-89-J-1063 of the Office of Naval Research. Research supported by MURST, Italy.     

Gap inequalities for non-convex mixed-integer quadratic programs

Laurent and Poljak introduced a very general class of valid linear inequalities, called gap inequalities, for the max-cut problem. We show that an analogous class of inequalities can be defined for general non-convex mixed-integer quadratic programs. These inequalities dominate some inequalities arising from a natural semidefinite relaxation.     

A note on the continuous mixing set

The continuous mixing set is , where w₁,…,w_n>0 and f₁,…,f_n∈ℜ. Let m=|{w₁,…,w_n}|. We show that when w₁|?|w_n, optimization over S can be performed in time O(n^m+1), and in time O(nlogn) when w₁=?=w_n=1.     

On the orthogonal Latin squares polytope

Since 1782, when Euler addressed the question of existence of a pair of orthogonal Latin squares (OLS) by stating his famous conjecture, these structures have remained an active area of research. In this paper, we examine the polyhedral aspects of OLS. In particular, we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. Two of these classes are shown to induce facet-defining inequalities of Chvátal rank two. For each such class, we provide a polynomial separation algorithm of the lowest possible complexity.