Clique facets of the axial and planar assignment polytopes (original) (raw)

The Facial Structure of the Clique Partitioning Polytope

The clique partitioning problem CPP can be formulated as follows. Given is a complete graph G = V;E, with edge weights w ij 2 R for all fi; jg 2 E . A subset A E is called a clique partition if there is a partition of V into non-empty, disjoint sets V 1 ; : : : ; V k , such that each V p p = 1 ; : : : ; k induces a clique i.e. a complete subgraph, and A = S k p=1 ffi; jgji; j 2 V p g. The weight of such a clique partition A is de ned as P fi;jg2A w ij . The problem is now to nd a clique partition of maximal weight. The clique partitioning polytope P is the convex hull of the incidence vectors of all clique partitions of G. In this paper we i n troduce several new classes of facet de ning inequalities of the clique partitioning polytope, and we present procedures that combine facet de ning inequalities into new ones. Finally, w e c haracterize all facet de ning inequalities with right hand side equal to 1 or 2.

Facets of the clique partitioning polytope

Mathematical Programming, 1990

is the convex hull of the incidence vectors of the clique partitionings of K~. We show that triangles, 2-chorded odd cycles, 2-chorded even wheels and other subgraphs of K,, induce facets of ~,,. The theoretical results described here have been used to design an (empirically) efficient cutting plane algorithm with which large (real-world) instances of the clique partitioning problem could be solved. These computational results can be found in GriStschel and Wakabayashi (1989).

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.

Facets of the three-index assignment polytope

Discrete Applied Mathematics, 1989

Given three disjoint n-sets and the family of all weighted triplets that contain exactly one element of each set, the 3-index assignment (or 3-dimensional matching) problem asks for a minimum-weight subcollection of triplets that covers exactly (i.e., partitions) the union of the three sets. Unlike the common (tindex) assignment problem, the 3-index problem is NPcomplete. In this paper we examine the facial structure of the 3-index assignment polytope (the convex hull of feasible solutions to the problem) with the aid of the intersection graph of the coefficient matrix of the problem's constraint set. In particular, we describe the cliques of the intersection graph as belonging to three distinct classes, and show that cliques in two of the three classes induce inequalities that define facets of our polytope. Furthermore, we give an O(n4) procedure (note that the number of variables is n3) for finding a facet-defining clique-inequality violated by a given noninteger solution to the linear programming relaxation of the 3-index assignment problem, or showing that no such inequality exists. We then describe the odd holes of the intersection graph and identify two classes of facets associated with odd holes that are easy to generate. One class has coefficients of 0 or 1, the other class coefficients of 0, 1 or 2. No odd hole inequality has left-hand side coefficients greater than two.

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.

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.

Some new classes of facets for the equicut polytope

Discrete Applied Mathematics, 1995

Given a graph G = (Y, E), a cut in G that partitions I/' into two sets with L i 1 V 1 J and r f 1 VI 1 nodes is called an equicut. Suppose that there are weights assigned to the edges in E. The problem of finding a minimum weight equicut in G is known to be NP-hard. The equicut polytope is defined as the convex hull of the incidence vectors of the equicuts in G. In this paper we describe several new classes of facets for the equicut polytope; they arise as various generalizations of an inequality based on a cycle introduced by Conforti et al. (1990). Most of our inequalities have the interesting feature that their support graphs are planar but for some of them both planarity and connectivity properties are lost. Finally we show how our results can be applied to obtain new classes of facets for the cut polytope.

On the Graph Bisection Cut Polytope

SIAM Journal on Discrete Mathematics, 2008

Given a graph G = (V, E) with node weights ϕ v ∈ N ∪ {0}, v ∈ V , and some number F ∈ N∪{0}, the convex hull of the incidence vectors of all cuts δ(S), S ⊆ V with ϕ(S) ≤ F and ϕ(V \ S) ≤ F is called the bisection cut polytope. We study the facial structure of this polytope which shows up in many graph partitioning problems with applications in VLSI-design or frequency assignment. We give necessary and in some cases sufficient conditions for the knapsack tree inequalities introduced in [9] to be facet-defining. We extend these inequalities to a richer class by exploiting that each cut intersects each cycle in an even number of edges. Finally, we present a new class of inequalities that are based on non-connected substructures yielding non-linear right-hand sides. We show that the supporting hyperplanes of the convex envelope of this non-linear function correspond to the faces of the so-called cluster weight polytope, for which we give a complete description under certain conditions.

Proving Facetness of Valid Inequalities for the Clique Partitioning Polytope

In this paper we prove two lifting theorems for the clique partitioning problem. Each of these theorems implies that if a valid inequality satis es certain conditions, then it de nes a facet of the clique partitioning polytope. In particular if a valid inequality de nes a facet of the polytope corresponding to the graph K m , i.e. the complete graph on m vertices, it de nes a facet for the polytope corresponding to K n for all n > m. This answers a question raised by Gr otschel and Wakabayashi.

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.