Facet-defining Inequalities Research Articles

On a generalization of the master cyclic group polyhedron

We study the master equality polyhedron (MEP) which generalizes the master cyclic group polyhedron (MCGP) and the master knapsack polyhedron (MKP). We present an explicit characterization of the polar of the nontrivial facet-defining inequalities for MEP. This result generalizes similar results for the MCGP by Gomory (1969) and for the MKP by Araoz (1974). Furthermore, this characterization gives a polynomial time algorithm for separating an arbitrary point from MEP. We describe how facet-defining inequalities for the MCGP can be lifted to obtain facet-defining inequalities for MEP, and also present facet-defining inequalities for MEP that cannot be obtained in such a way. Finally, we study the mixed-integer extension of MEP and present an interpolation theorem that produces valid inequalities for general mixed integer programming problems using facets of MEP.

Weighted graphs defining facets: A connection between stable set and linear ordering polytopes

A graph is α -critical if its stability number increases whenever an edge is removed from its edge set. The class of α -critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lovász [L. Lovász, Some finite basis theorems on graph theory, in: A. Hajnal, V.T. Sós (Eds.), Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, in: Colloquia Mathematica Societatis János Bolyai, vol. 18, North-Holland, Amsterdam, 1978, pp. 717–729] is the finite basis theorem for α -critical graphs of a fixed defect. The class of α -critical graphs is also of interest for at least two topics of polyhedral studies. First, Chvátal [V. Chvátal, On certain polytopes associated with graphs, Journal of Combinatorial Theory Ser. B 18 (1975) 138–154] shows that each α -critical graph induces a rank inequality which is facet-defining for its stable set polytope. Investigating a weighted generalization, Lipták and Lovász [L. Lipták, L. Lovász, Facets with fixed defect of the stable set polytope, Mathematical Programming 88 (Ser. A) (2000) 33–44; L. Lipták, L. Lovász, Critical facets of the stable set polytope, Combinatorica 21 (2001) 61–88] introduce critical facet-graphs (which again produce facet-defining inequalities for their stable set polytopes) and they establish a finite basis theorem. Second, Koppen [M. Koppen, Random utility representation of binary choice probabilities: Critical graphs yielding critical necessary conditions, Journal of Mathematical Psychology 39 (1995) 21–39] describes a construction that delivers from any α -critical graph a facet-defining inequality for the linear ordering polytope. Doignon et al. [J.-P. Doignon, S. Fiorini, G. Joret, Facets of the linear ordering polytope: A unification for the fence family through weighted graphs, Journal of Mathematical Psychology 50 (3) (2006) 251–262] handle the weighted case and thus define facet-defining graphs. Here we investigate relationships between the two weighted generalizations of α -critical graphs. We show that facet-defining graphs (for the linear ordering polytope) are obtainable from 1-critical facet-graphs (linked with stable set polytopes). We then use this connection to derive various results on facet-defining graphs, the most prominent one being derived from Lipták and Lovász’s finite basis theorem for critical facet-graphs. At the end of the paper we offer an alternative proof of Lovász’s finite basis theorem for α -critical graphs.

Lifting inequalities: a framework for generating strong cuts for nonlinear programs

In this paper, we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs. The theory is geometric and provides insights into lifting-based cut generation procedures, yielding short proofs of earlier results in mixed-integer programming. Using convex extensions, we obtain conditions that allow for sequence-independent lifting in nonlinear settings, paving a way for efficient cut-generation procedures for nonlinear programs. This sequence-independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general-purpose, strong cuts for integer programs. We specialize our lifting results to derive facet-defining inequalities for mixed-integer bilinear knapsack sets. Finally, we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut-generation techniques applied on a tight reformulation of the problem.

Facets of Two-Dimensional Infinite Group Problems

In this paper, we lay the foundation for the study of the two-dimensional mixed integer infinite group problem (2DMIIGP). We introduce tools to determine if a given continuous and piecewise linear function over the two-dimensional infinite group is subadditive and to determine whether it defines a facet of 2DMIIGP. We then present two different constructions that yield the first known families of facet-defining inequalities for 2DMIIGP. The first construction uses valid inequalities of the one-dimensional integer infinite group problem (1DIIGP) as building blocks for creating inequalities for the two-dimensional integer infinite group problem (2DIIGP). We prove that this construction yields all continuous piecewise linear facets of the two-dimensional group problem that have exactly two gradients. The second construction we present has three gradients and yields facet-defining inequalities of 2DMIIGP whose continuous coefficients are not dominated by those of facets of the one-dimensional mixed integer infinite group problem (1DMIIGP).

On the Polytope of the (1,2)-Survivable Network Design Problem

This paper deals with the survivable network design problem where each node v has a connectivity type $r(v)$ equal to 1 or 2, and the survivability conditions require the existence of at least $\min\{r(s),r(t)\}$ edge-disjoint paths for all distinct nodes s and t. We consider the polytope given by the trivial and cut inequalities together with the partition inequalities. More precisely, we study some structural properties of this polytope which leads us to give some sufficient conditions for this polytope to be integer in the class of series-parallel graphs. With both separation problems for the cut and partition inequalities being polynomially solvable, we then obtain a polynomial time algorithm for the (1,2)-survivable network design problem in a subclass of series-parallel graphs including the outerplanar graph class. We also introduce a new class of facet-defining inequalities for the polytope associated to the (1,2)-survivable network design problem.

Strong valid inequalities for the resource-constrained scheduling problem with uniform resource requirements

We consider the resource-constrained scheduling problem when each job’s resource requirements remain constant over its processing time. We study a time-indexed formulation of the problem, providing facet-defining inequalities for a projection of the resulting polyhedron that exploit the resource limitations inherent in the problem. Lifting procedures are then provided for obtaining strong valid inequalities for the original polyhedron. Computational results are presented to demonstrate the strength of these inequalities.

Simultaneously lifting sets of binary variables into cover inequalities for knapsack polytopes

Cover inequalities are commonly used cutting planes for the 0–1 knapsack problem. This paper describes a linear-time algorithm (assuming the knapsack is sorted) to simultaneously lift a set of variables into a cover inequality. Conditions for this process to result in valid and facet-defining inequalities are presented. In many instances, the resulting simultaneously lifted cover inequality cannot be obtained by sequentially lifting over any cover inequality. Some computational results demonstrate that simultaneously lifted cover inequalities are plentiful, easy to find and can be computationally beneficial.

Packing and partitioning orbitopes

We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at most or exactly one 1-entry in each row, respectively. The goal of investigating these polytopes is to gain insight into ways of breaking certain symmetries in integer programs by adding constraints, e.g., for a well-known formulation of the graph coloring problem. We provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns is the cyclic group or the symmetric group. Our main results are complete linear inequality descriptions of these polytopes by facet-defining inequalities. For the cyclic group case, the descriptions turn out to be totally unimodular, while for the symmetric group case, both the description and the proof are more involved. The associated separation problems can be solved in linear time.

The Symmetric Traveling Salesman Polytope: New Facets from the Graphical Relaxation

The path, the wheelbarrow, and the bicycle inequalities have been shown by Cornuéjols, Fonlupt, and Naddef to be facet-defining for the graphical relaxation of STSP(n), the polytope of the symmetric traveling salesman problem on an n-node complete graph. We show that these inequalities, and some generalizations of them, define facets also for STSP(n). In conclusion, we characterize a large family of facet-defining inequalities for STSP(n) that include, as special cases, most of the inequalities currently known to have this property as the comb, the clique tree, and the chain inequalities. Most of the results given here come from a strong relationship of STSP(n) with its graphical relaxation that we have pointed out in another paper, where the basic proof techniques are also described.

The Integer Knapsack Cover Polyhedron

We study the integer knapsack cover polyhedron which is the convex hull of the set of vectors $x\in \mathbb{Z}_{+}^{n}$ that satisfy $C^{T}x\geq b$, with $C\in \mathbb{Z}_{++}^{n}$ and $b\in \mathbb{Z}_{++}$. We present some general results about the nontrivial facet-defining inequalities. Then we derive specific families of valid inequalities, namely, rounding, residual capacity, and lifted rounding inequalities, and identify cases where they define facets. We also study some known families of valid inequalities called 2-partition inequalities and improve them using sequence-independent lifting.

Facet-defining inequalities for the simple graph partitioning polytope

The simple graph partitioning problem is to partition an edge-weighted graph into mutually node-disjoint subgraphs, each containing at most b nodes, such that the sum of the weights of all edges in the subgraphs is maximal. In this paper we provide several classes of facet-defining inequalities for the associated simple graph partitioning polytope.

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.

Generalized clique family inequalities for claw-free graphs

Abstract Providing a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem [M. Grotschel, L. Lovasz, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988]. For the subclass of quasi-line graphs, Ben Rebea conjectured and Eisenbrandt et al. [F. Eisenbrand, G. Oriolo, G. Stauffer, P. Ventura, Circular Ones Matrices and the Stable Set Polytope of Quasi-Line Graphs, in: Lecture Notes in Computer Science, 3509, 2005, pp. 291–305] recently proved that all non-trivial facets belong to only one class, the so-called clique family inequalities. For general claw-free graphs, however, more complex facets are required to describe the stable set polytope. We introduce a generalization of clique family inequalities, and prove that several facet-defining inequalities for general claw-free graphs are of this type.

Multi-period capacitated location with modular equipments

This work deals with a multi-period capacitated location problem inspired by telecommunication access network planning problems, where demands and costs vary from one period to another. On each concentrator site, several capacitated concentrators can be installed at each period. Similarly, several capacitated modules can be installed at each period between each terminal and concentrator sites. We assume that equipments can never be removed. An integer linear model is proposed, and some of its dynamic properties are investigated. Then, a polyhedral analysis of the problem is performed, and some original facet-defining inequalities are introduced. The different improvements proposed are validated on numerical examples.

Description of 2-integer continuous knapsack polyhedra

In this paper we discuss the polyhedral structure of several mixed integer sets involving two integer variables. We show that the number of the corresponding facet-defining inequalities is polynomial on the size of the input data and their coefficients can also be computed in polynomial time using a known algorithm [D. Hirschberg, C. Wong, A polynomial-time algorithm for the knapsack problem with two variables, Journal of the Association for Computing Machinery 23 (1) (1976) 147–154] for the two integer knapsack problem. These mixed integer sets may arise as substructures of more complex mixed integer sets that model the feasible solutions of real application problems.

Facets of the linear ordering polytope: A unification for the fence family through weighted graphs

The binary choice polytope appeared in the investigation of the binary choice problem formulated by Guilbaud and Block and Marschak. It is nowadays known to be the same as the linear ordering polytope from operations research (as studied by Grötschel, Jünger and Reinelt). The central problem is to find facet-defining linear inequalities for the polytope. Fence inequalities constitute a prominent class of such inequalities (Cohen and Falmagne; Grötschel, Jünger and Reinelt). Two different generalizations exist for this class: the reinforced fence inequalities of Leung and Lee, and independently Suck, and the stability–critical fence inequalities of Koppen. Together with the fence inequalities, these inequalities form the fence family. Building on previous work on the biorder polytope by Christophe, Doignon and Fiorini, we provide a new class of inequalities which unifies all inequalities from the fence family. The proof is based on a projection of polytopes. The new class of facet-defining inequalities is related to a specific class of weighted graphs, whose definition relies on a curious extension of the stability number. We investigate this class of weighted graphs which generalize the stability–critical graphs.

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.

Lot Sizing with Inventory Bounds and Fixed Costs: Polyhedral Study and Computation

We investigate the polyhedral structure of the lot-sizing problem with inventory bounds. We consider two models, one with linear cost on inventory, the other with linear and fixed costs on inventory. For both models, we identify facet-defining inequalities that make use of the inventory bounds explicitly and give exact separation algorithms. We also describe a linear programming formulation of the problem when the order and inventory costs satisfy the Wagner-Whitin nonspeculative property. We present computational experiments that show the effectiveness of the results in tightening the linear programming relaxations of the lot-sizing problem with inventory bounds and fixed costs.

New Results on the Mixed General Routing Problem

In this paper, we deal with the polyhedral description and the resolution of the Mixed General Routing Problem. This problem, in which the service activity occurs both at some of the nodes and at some of the arcs and edges of a mixed graph, contains a large number of important arc and node routing problems as special cases. Here, a large family of facet-defining inequalities, the Honeycomb inequalities, is described. Furthermore, a cutting-plane algorithm for this problem that incorporates new separation procedures for the K-C, Regular Path-Bridge, and Honeycomb inequalities is presented. Branch and bound is invoked when the final solution of the cutting-plane procedure is fractional. Extensive computational experiments over different sets of instances are included.

Transformation of Facets of the General Routing Problem Polytope

We study the class of polytopes associated with the 0/1-variable formulation by [G. Ghiani and G. Laporte, Math. Program., 87 (2000), pp. 467--481] of the rural postman problem and general routing problem. We introduce a technique called switching, which allows one to obtain new valid or facet-defining inequalities from known ones. We study the switched forms which can be obtained from two important types of known valid inequalities, namely, so called path-bridge [A. N. Letchford, Eur. J. Oper. Res., 96 (1997), pp. 317--322] and honeycomb [A. Corberan and J. M. Sanchis, Eur. J. Oper. Res., 108 (1998), pp. 538--550] inequalities. For the switched path-bridge inequalities, a generic blueprint for separation routines is developed. We give a polynomial time separation algorithm for a special subclass of these inequalities.