Facets of the three-index assignment polytope (original) (raw)

On facets of the three-index assignment polytope

1992

Two new classes of facet-defining inequalities for the three-index assignrnent polytope are identified in this paper. According to the shapes of their support index sets, we call these facets bull facets and comb facets respectively. The bull facet has Chvatal rank 1, while the comb facet has Chvatal rank 2. For a comb facet-defining inequality, the right-hand-side coefficient is a positive integer, and the left-ha,nd-side coefficients equal to 0 or 1. For a bull facet-defining inequality, the right-hand-side coefficient is a positive even integer, and the left-hand-side coefficients equal to 0, 1 or 2. Furthermore, we give an O(n 3) procedure for finding a bull facet with the right-hand-side coefficient 2, violated by a given noninteger solution to the linear programming relaxation of the three-index assignment problem, or showing that no such facet exists. Such an algorithm is called a separation algorithm. Since the number of variables is n 3 and one needs to check through all the variables in such a separation algorithm, this algorithm is linear-time and the order of its complexity is the best possible.

On multi-index assignment polytopes

2006

We investigate an integer programming model for multi-dimensional assignment problems. This model enables us to establish the dimension for entire families of assignment polytopes, thus unifying and generalising previous results. In particular, we establish the dimension of the linear assignment polytope as well as that of every axial and planar assignment polytope. Further, for the axial polytopes, we identify a family of clique facets. We also give a necessary condition for the existence of a solution for assignment problems.

Clique facets of the axial and planar assignment polytopes

Discrete Optimization, 2009

The (k, s) assignment problem sets a unified framework for studying the facial structure of families of assignment polytopes. Through this framework, we derive classes of clique facets for all axial and planar assignment polytopes. For each of these classes, a polynomialtime separation procedure is described. Furthermore, we provide computational experience illustrating the efficiency of these facet-defining inequalities when applied as cutting planes.

On the complete set packing and set partitioning polytopes: Properties and rank 1 facets

Operations Research Letters, 2018

This paper studies two polytopes: the complete set packing and set partitioning polytopes, which are both associated with a binary n-row matrix having all possible columns. Cuts of rank 1 for the latter polytope play a central role in recent exact algorithms for many combinatorial problems, such as vehicle routing. We show the precise relation between the two polytopes studied, characterize the multipliers that induce rank 1 clique facets and give several families of multipliers that yield other facets.

Polyhedral results for assignment problems

2002

This paper introduces an Integer Programming model for multidimensional assignment problems and examines the underlying polytopes. It also proposes a certain hierarchy among assignment polytopes. The dimension for classes of multidimensional assignment polytopes is established, unifying and generalising previous results. The framework introduced constitutes the first step towards a polyhedral characterisation for classes of assignment problems. The generic nature of this approach is illustrated by identifying a family of facets for a certain class of multidimensional assignment problems, namely "axial" problems. disjoint tuples, each including a single element from each set. This is the class of axial assignment problems ([3, 28]). A different structure appears, if the aim is, instead, to identify a collection of ¦ © tuples, partitioned into ¦ disjoint sets of ¦ disjoint tuples. By way of illustration, consider the problem of allocating ¦ teachers to ¦ student groups for sessions in one of ¦ classrooms and using one of ¦ laboratory facilities, in such a way that all teachers teach all groups, using each time a different classroom or a different facility (for a relevant case, see [11]). These assignment problems are called planar and are directly linked to structures called Mutually Orthogonal Latin Squares (MOLS) ([17]). Generalising this concept, we could ask for an optimal collection of ¦ tuples, thus defining the ¥ assignment problem of order ¦ hereafter denoted as ¥ A P , which encompasses all known assignment structures.

A new class of facets for the Latin square polytope

Discrete Applied Mathematics, 2006

Latin squares of order n have a 1-1 correspondence with the feasible solutions of the 3-index planar assignment problem (3PAP n). In this paper, we present a new class of facets for the associated polytope, induced by odd-hole inequalities.

Total Coloring and Total Matching: Polyhedra and Facets

European Journal of Operational Research, 2022

A total coloring of a graph G = (V, E) is an assignment of colors to vertices and edges such that neither two adjacent vertices nor two incident edges get the same color, and, for each edge, the end-points and the edge itself receive different colors. Any valid total coloring induces a partition of the elements of G into total matchings, which are defined as subsets of vertices and edges that can take the same color. In this paper, we propose Integer Linear Programming models for both the Total Coloring and the Total Matching problems, and we study the strength of the corresponding Linear Programming relaxations. The total coloring is formulated as the problem of finding the minimum number of total matchings that cover all the graph elements. This covering formulation can be solved by a Column Generation algorithm, where the pricing subproblem corresponds to the Weighted Total Matching Problem. Hence, we study the Total Matching Polytope. We introduce three families of nontrivial valid inequalities: vertex-clique inequalities based on standard clique inequalities of the Stable Set Polytope, congruent-2k3 cycle inequalities based on the parity of the vertex set induced by the cycle, and even-clique inequalities induced by complete subgraphs of even order. We prove that congruent-2k3 cycle inequalities are facet-defining only when k = 4 , while the vertex-clique and even-cliques are always facet-defining. Finally, we present preliminary computational results of a Column Generation algorithm for the Total Coloring Problem and a Cutting Plane algorithm for the Total Matching Problem.

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

Discrete Optimization, 2009

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.

On the 0, 1 facets of the set covering polytope

Mathematical Programming, 1989

In this paper,j we consider: inequalities of t-G form ",.-equals 0 or 1, and is a positive integer. We give necessary and sufficient conditions for ... such inequalities to define facets of the set covering polytope associated to a 0,1 constraint matrix A. These conditions are in terms of critical edges and critical cutsets defined in the bipartite incidence graph associated to A, and are very much in the spirit of the work of Balas and Zemel on the set packing problem where similar notions were defined in the intersection graph of A. Furthermore, we give a polynomial characterization of a class of 0,1 facets defined from chorded cycles induced in the bipartite incidence graph. This characterization also yields all the 0,1 liftings of odd-hole inequalities for the simple plant location polytope.

On the set covering polytope:Facets with coefficients in {0, 1, 2, 3}

Annals of Operations Research, 1998

Balas and Ng [1, 2] characterized the class of valid inequalities for the set coveringpolytope with coefficients equal to 0, 1 or 2, and gave necessary and sufficient conditionsfor such an inequality to be facet defining. We extend this study, characterizing the class ofvalid inequalities with coefficients equal to 0, 1, 2 or 3, and giving necessary and sufficientconditions for