On Facets of Knapsack Equality Polytopes

The 0/1 knapsack equality polytope is, by definition, the convex hull of 0/1 solutions of a single linear equation. A special form of this polytope, where the defining linear equation has nonnegative integer coefficients and the number of variables having coefficient one exceeds the right-hand side, is considered. Equality constraints of this form arose in a real-world application of integer programming to a truck dispatching scheduling problem. Families of facet defining inequalities for this polytope are identified, and complete linear inequality representations are obtained for some classes of polytopes.     

Size-constrained graph partitioning polytopes

We consider the problem of clustering a set of items into subsets whose sizes are bounded from above and below. We formulate the problem as a graph partitioning problem and propose an integer programming model for solving it. This formulation generalizes several well-known graph partitioning problems from the literature like the clique partitioning problem, the equi-partition problem and the k-way equi-partition problem. In this paper, we analyze the structure of the corresponding polytope and prove several results concerning the facial structure. Our analysis yields important results for the closely related equi-partition and k-way equi-partition polytopes as well.     

Ramsey theory and integrality gap for the independent set problem

We discuss the effectiveness of integer programming for solving large instances of the independent set problem. Typical LP formulations, even strengthened by clique inequalities, yield poor bounds for this problem. We show that a strong bound can be obtained by the use of the so-called rank inequalities, which generalize the clique inequalities. For some problems the clique inequalities imply the rank inequalities, and then a strong bound is guaranteed already by the simpler formulation.     

Valid inequalities and facets of the capacitated plant location problem

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure.The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with a constant capacity for all plants. These facet inequalities depend on the capacity and thus differ fundamentally from the valid inequalities for the uncapacited version of the problem.We also introduce a second formulation for a model with indivisible customer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.This research was partially supported by Grant # ECS-8316224 from the National Science Foundation's Program in Systems Theory and Operations Research.     

A study of the quadratic semi-assignment polytope

We study a polytope which arises from a mixed integer programming formulation of the quadratic semi-assignment problem. We introduce an isomorphic projection and transform the polytope to a tractable full-dimensional polytope. As a result, some basic polyhedral properties, such as the dimension, the affine hull, and the trivial facets, are obtained. Further, we present valid inequalities called cut- and clique-inequalities and give complete characterizations for them to be facet-defining. We also discuss a simultaneous lifting of the clique-type facets. Finally, we show an application of the quadratic semi-assignment problem to hub location problems with some computational experiences.     

Polyhedral studies on the convex recoloring problem

A coloring of the vertices of a graph G is convex if, for each assigned color d, the vertices with color d induce a connected subgraph of G. We address the convex recoloring problem, defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G, so that the resulting coloring is convex. This problem is known to be NP-hard even when G is a path. We show an integer programming formulation for the weighted version of this problem on arbitrary graphs, and then specialize it for trees. We study the facial structure of the polytope defined as the convex hull of the integer points satisfying the restrictions of the proposed ILP formulation, present several classes of facet-defining inequalities and discuss separation algorithms.     

A polyhedral study of the maximum edge subgraph problem

The study of cohesive subgroups is an important aspect of social network analysis. Cohesive subgroups are studied using different relaxations of the notion of clique in a graph. For instance, given a graph and an integer k, the maximum edge subgraph problem consists in finding a k-vertex subset such that the number of edges within the subset is maximum. This work proposes a polyhedral approach for this NP-hard problem. We study the polytope associated to an integer programming formulation of the problem, present several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families.     

A time indexed formulation of non-preemptive single machine scheduling problems

We consider the formulation of non-preemptive single machine scheduling problems using time-indexed variables. This approach leads to very large models, but gives better lower bounds than other mixed integer programming formulations. We derive a variety of valid inequalities, and show the role of constraint aggregation and the knapsack problem with generalised upper bound constraints as a way of generating such inequalities. A cutting plane/branch-and-bound algorithm based on these inequalities has been implemented. Computational experience on small problems with 20/30 jobs and various constraints and objective functions is presented.The research of this author was partially supported by JNICT/INVOTAN under grant No. 30/A/85/PO and by the PAC, contract No. 87/92-106, for computation.     

An interior point algorithm to solve computationally difficult set covering problems

We present an interior point approach to the zero–one integer programming feasibility problem based on the minimization of a nonconvex potential function. Given a polytope defined by a set of linear inequalities, this procedure generates a sequence of strict interior points of this polytope, such that each consecutive point reduces the value of the potential function. An integer solution (not necessarily feasible) is generated at each iteration by a rounding scheme. The direction used to determine the new iterate is computed by solving a nonconvex quadratic program on an ellipsoid. We illustrate the approach by considering a class of difficult set covering problems that arise from computing the 1-width of the incidence matrix of Steiner triple systems.     

Partial linear characterizations of the asymmetric travelling salesman polytope

We consider the linear programming formulation of the asymmetric travelling salesman problem. Several new inequalities are stated which yield a sharper characterization in terms of linear inequalities of the travelling salesman polytope, i.e., the convex hull of tours. In fact, some of the new inequalities as well as some of the well-known subtour elimination constraints are indeed facets of the travelling salesman polytope, i.e., belong to the class of inequalities that uniquely characterize the convex hull of tours to an-city problem.     

The simple plant location problem: Survey and synthesis

With emphasis on the simple plant location problem, (SPLP), we consider an important family of discrete, deterministic, single-criterion, NP-hard, and widely applicable optimization problems. The introductory discussion on problem formulation aspects is followed by the establishment of relationships between SPLP and set packing, set covering and set partitioning problems which all are among those structures in integer programming having the most wide-spread applications. An extensive discourse on solution properties and computational techniques, spanning from early heuristics to the presumably most novel exact methods is then provided. Other subjects of concern include a subfamily of SPLP's solvable in polynomial time, analyses of approximate algorithms, transformability of p-CENTER and p-MEDIAN to SPLP, and structural properties of the SPLP polytope. Along the way we attempt to synthesize these findings and relate them to other areas of integer programming.     

On the set covering polytope: I. All the facets with coefficients in {0, 1, 2}

While the set packing polytope, through its connection with vertex packing, has lent itself to fruitful investigations, little is known about the set covering polytope. We characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0, 1 or 2, and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining. We show that all inequalities in the above class are contained in the elementary closure of the constraint set, and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure. We point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix. Finally, we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation.Research supported by the National Science Foundation through grant ECS-8503198 and the Office of Naval Research through contract N0001485-K-0198.     

A branch-and-cut algorithm for the Multiple Steiner TSP with Order constraints

The paper deals with a problem motivated by survivability issues in multilayer IP-over-WDM telecommunication networks. Given a set of traffic demands for which we know a survivable routing in the IP layer, our purpose is to look for the corresponding survivable topology in the WDM layer. The problem amounts to Multiple Steiner TSPs with order constraints. We propose an integer linear programming formulation for the problem and investigate the associated polytope. We also present new valid inequalities and discuss their facial aspect. Based on this, we devise a Branch-and-cut algorithm and present preliminary computational results.     

A tree search algorithm for the crew scheduling problem

In this paper we consider the crew scheduling program, that is the problem of assigning K crews to N tasks with fixed start and finish times such that each crew does not exceed a limit on the total time it can spend working.A zero-one integer linear programming formulation of the problem is given, which is then relaxed in a lagrangean way to provide a lower bound which is improved by subgradient optimisation. Finally a tree search procedure incorporating this lower bound is presented to solve the problem to optimality.Computational results are given for a number of randomly generated test problems involving between 50 and 500 tasks.     

Polyhedral results for the edge capacity polytope

Network loading problems occur in the design of telecommunication networks, in many different settings. For instance, bifurcated or non-bifurcated routing (also called splittable and unsplittable) can be considered. In most settings, the same polyhedral structures return. A better understanding of these structures therefore can have a major impact on the tractability of polyhedral-guided solution methods. In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case.?As an example, one of the basic variants of network loading is described, including an integer linear programming formulation. As the edge capacity problems are relaxations of this network loading problem, their polytopes are intimately related. We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope. We describe classes of strong valid inequalities for the edge capacity polytopes, and we derive conditions under which these constraints define facets. For the diverse classes the complexity of lifting projected variables is stated.?The derived inequalities are tested on (i) the edge capacity problem itself and (ii) the described variant of the network loading problem. The results show that the inequalities substantially reduce the number of nodes needed in a branch-and-cut approach. Moreover, they show the importance of the edge subproblem for solving network loading problems. Received: September 2000 / Accepted: October 2001?Published online March 27, 2002     

On discrete lot-sizing and scheduling on identical parallel machines

We consider the multi-item discrete lot-sizing and scheduling problem on identical parallel machines. Based on the fact that the machines are identical, we introduce aggregate integer variables instead of individual variables for each machine. For the problem with start-up costs, we show that the inequalities based on a unit flow formulation for each machine can be replaced by a single integer flow formulation without any change in the resulting LP bound. For the resulting integer lot-sizing with start-ups subproblem, we show how inequalities for the unit demand case can be generalized and how an approximate version of the extended formulation of Eppen and Martin can be constructed. The results of some computational experiments carried out to compare the effectiveness of the various mixed-integer programming formulations are presented.     

Solving a multicoloring problem with overlaps using integer programming

This paper presents a new generalization of the graph multicoloring problem. We propose a Branch-and-Cut algorithm based on a new integer programming formulation. The cuts used are valid inequalities that we could identify to the polytope associated with the model. The Branch-and-Cut system includes separation heuristics for the valid inequalities, specific initial and primal heuristics, branching and pruning rules. We report on computational experience with random instances.     

Unrelated machine scheduling with time-window and machine downtime constraints: An application to a naval battle-group problem

This paper deals with an unrelated machine scheduling problem of minimizing the total weighted flow time, subject to time-window job availability and machine downtime side constraints. We present a zero-one integer programming formulation of this problem. The linear programming relaxation of this formulation affords a tight lower bound and often generates an integer optimal solution for the problem. By exploiting the special structures inherent in the formulation, we develop some classes of strong valid inequalities that can be used to tighten the initial formulation, as well as to provide cutting planes in the context of a branch-and-cut procedure. A major computational bottleneck is the solution of the underlying linear programming relaxation because of the extremely high degree of degeneracy inherent in the formulation. In order to overcome this difficulty, we employ a Lagrangian dual formulation to generate lower and upper bounds and to drive the branch-and-bound algorithm. As a practical instance of the unrelated machine scheduling problem, we describe a combinatorial naval defense problem. This problem seeks to schedule a set of illuminators (passive homing devices) in order to strike a given set of targets using surface-to-air missiles in a naval battle-group engagement scenario. We present computational results for this problem using suitable realistic data.     

Small Chvátal Rank

We propose a variant of the Chvátal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {x : A x ≤ b} as b varies. The number of steps needed is called the small Chvátal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube.     

Valid inequalities for a single constrained 0-1 MIP set intersected with a conflict graph

In this paper a mixed integer set resulting from the intersection of a single constrained mixed 0–1 set with the vertex packing set is investigated. This set arises as a subproblem of more general mixed integer problems such as inventory routing and facility location problems. Families of strong valid inequalities that take into account the structure of the simple mixed integer set and that of the vertex packing set simultaneously are introduced. In particular, the well-known mixed integer rounding inequality is generalized to the case where incompatibilities between binary variables are present. Exact and heuristic algorithms are designed to solve the separation problems associated to the proposed valid inequalities. Preliminary computational experiments show that these inequalities can be useful to reduce the integrality gaps and to solve integer programming problems.