On the Graph Bisection Cut Polytope (original) (raw)

Generating facets for the cut polytope of a graph by triangular elimination

Mathematical Programming, 2007

The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs. Triangular elimination is a specific combination of zero-lifting and Fourier-Motzkin elimination using the triangle inequality. We prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing. The proof is based on a variation of the lifting lemma adapted to general graphs. The result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph. We also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination. *

A Polynomial Time Algorithm for Vertex Enumeration and Optimization over Shaped Partition Polytopes

We consider the Shaped Partition Problem of partitioning n given vectors in real k-space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. In addressing this problem, we study the Shaped Partition Polytope defined as the convex hull of solutions. The Shaped Partition Problem captures N P-hard problems such as the Max-Cut problem and the Traveling Salesperson problem, and the Shaped Partition Polytope may have exponentially many vertices and facets, even when k or p are fixed. In contrast, we show that when both k and p are fixed, the number of vertices is polynomial in n, and all vertices can be enumerated and the optimization problem solved in strongly polynomial time. Explicitly, we show that any Shaped Partition Polytope has O(n k(p 2)) vertices which can be enumerated in O(n k 2 p 3) arithmetic operations, and that any Shaped Partition Problem is solvable in O(n kp 2) arithmetic operations.

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.

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 strength of cut-based inequalities for capacitated network design polyhedra

In this paper we study capacitated network design problems, differentiating directed, bidirected and undirected link capacity models. We complement existing polyhedral results for the three variants by new classes of facet-defining valid inequalities and unified lifting results. For this, we study the restriction of the problems to a cut of the network. First, we show that facets of the resulting cutset polyhedra translate into facets of the original network design polyhedra if the two subgraphs defined by the network cut are (strongly) connected. Second, we provide an analysis of the facial structure of cutset polyhedra, elaborating the differences caused by the three different types of capacity constraints. We present flow-cutset inequalities for all three models and show under which conditions these are facet-defining. We also state a new class of facets for the bidirected and undirected case and it is shown how to handle multiple capacity modules by mixed-integer rounding (MIR).

An procedure for identifying facets of the knapsack polytope

Operations Research Letters, 2003

An O(n log n) procedure is presented for obtaining facets of the knapsack polytope by lifting the inequalities induced by the extensions of strong minimal covers. The procedure does not require any sequential lifting of the inequalities. In contrast, it utilizes the information from the maximal cliques implied by the knapsack constraint for determining the combination of the lifting coe cients to generate each facet.

On Cut Polytopes and Graph Minors

2020

The max-cut problem is a fundamental and much-studied NP-hard combinatorial optimisation problem, with a wide range of applications. The associated polyhedron, called the cut polytope, has been studied in some depth. A classic result of Barahona and Mahjoub states that the cut polytope of a graph G is completely described by co-circuit inequalities if and only if G does not contain the graph K5 as a minor. We present some more results of this type, which may lead to new polynomially-solvable special cases of the max-cut problem.

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.

Gap Inequalities for the Cut Polytope

European Journal of Combinatorics, 1996

We introduce a new class of inequalities valid for the cut polytope, which we call gap inequalities. Each gap inequality is given by a nite sequence of integers, whose \gap" is de ned as the smallest discrepancy arising when decomposing the sequence into two parts as equal as possible. Gap inequalities include the hypermetric inequalities and the negative type inequalities, which have been extensively studied in the literature. They are also related to a positive semide nite relaxation of the max-cut problem. A natural question is to decide for which integer sequences the corresponding gap inequalities de ne facets of the cut polytope. For this property, we present a set of necessary and su cient conditions in terms of the root patterns and of the rank of an associated matrix. We also prove that there is no facet de ning inequality with gap greater than one and which is induced by a sequence of integers using only two distinct values.

Facet-inducing inequalities with acyclic supports for the caterpillar-packing polytope

RAIRO - Operations Research, 2019

A caterpillar is a connected graph such that the removal of all its vertices with degree 1 results in a path. Given a graph G, a caterpillar-packing of G is a set of vertex-disjoint (not necessarily induced) subgraphs of G such that each subgraph is a caterpillar. In this work we consider the set of caterpillar-packings of a graph, which corresponds to feasible solutions of the 2-schemes strip cutting problem with a sequencing constraint (2-SSCPsc) presented by Rinaldi and Franz (Eur. J. Oper. Res. 183 (2007) 1371–1384). Facet-preserving procedures have been shown to be quite effective at explaining the facet-inducing inequalities of the associated polytope, so in this work we continue this issue by exploring such procedures for valid inequalities with acyclic supports. In particular, the obtained results are applicable when the underlying graph is a tree.