Fixed cardinality stable sets

Given an undirected graph G=(V,E) and a positive integer k in {1, ..., |V|}, we initiate the combinatorial study of stable sets of cardinality exactly k in G. Our aim is to instigate the polyhedral investigation of the convex hull of fixed cardinality stable sets, inspired by the rich theory on the classical structure of stable sets. We introduce a large class of valid inequalities to the natural integer programming formulation of the problem. We also present simple combinatorial relaxations based on computing maximum weighted matchings, which yield dual bounds towards finding minimum-weight fixed cardinality stable sets, and particular cases which are solvable in polynomial time.publishedVersio

The maximization of submodular functions : old and new proofs for the correctness of the dichotomy algorithm

The first purpose of this paper is to make an old (Russian) theoretical results about the structure of local and global maxima of submodular functions, Cherenin’s excluding rules and his Dichotomy Algorithm more accessible for Western community. The second purpose of this paper is to present our main result which can be stated as follows. For any pair of embedded subsets, the difference of their function values is a lower bound for the difference between the unknown(!) optimal values of the corresponding partition defined by these subsets. A simple justification of Cherenin’s rules, the Dichotomy Algorithmand its generalization with the new branching rules from our main result are presented. The usefulness of our new branching rules is illustrated by means of a numerical example.

Transitive Packing: A Unifying Concept in Combinatorial Optimization

This paper attempts to give a better understanding of the facial structure of previously separately investigated polyhedra. It introduces the notion of transitive packing and the transitive packing polytope. Polytopes that turn out to be special cases of the transitive packing polytope are, among others, the node packing polytope, the acyclic subdigraph polytope, the bipartite subgraph polytope, the planar subgraph polytope, the clique partitioning polytope, the partition polytope, the transitive acyclic subdigraph polytope, the interval order polytope, and the relatively transitive subgraph polytope. We give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope,in this way introducing generalized cycle, generalized clique, generalized antihole, generalized antiweb, and odd partition inequalities. These classes subsume several known classes of valid inequalities for several of the special cases and give also many new inequalities for several other special cases. For some of the classes we also prove a lower bound for their Gomory-Chvdtal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering as well as to balanced and ideal matrices

Polyhedral Approaches to Hypergraph Partitioning and Cell Formation

Ankara : Department of Industrial Engineering and Institute of Engineering and Science, Bilkent University, 1994.Thesis (Ph.D.) -- -Bilkent University, 1994.Includes bibliographical references leaves 152-161Hypergraphs are generalizations of graphs in the sense that each hyperedge can connect more than two vertices. Hypergraphs are used to describe manufacturing environments and electrical circuits. Hypergraph partitioning in manufacturing models cell formation in Cellular Manufacturing systems. Moreover, hypergraph partitioning in VTSI design case is necessary to simplify the layout problem. There are various heuristic techniques for obtaining non-optimal hypergraph partitionings reported in the literature. In this dissertation research, optimal seeking hypergraph partitioning approaches are attacked from polyhedral combinatorics viewpoint. There are two polytopes defined on r-uniform hypergraphs in which every hyperedge has exactly r end points, in order to analyze partitioning related problems. Their dimensions, valid inequality families, facet defining inequalities are investigated, and experimented via random test problems. Cell formation is the first stage in designing Cellular Manufacturing systems. There are two new cell formation techniques based on combinatorial optimization principles. One uses graph approximation, creation of a flow equivalent tree by successively solving maximum flow problems and a search routine. The other uses the polynomially solvable special case of the one of the previously discussed polytopes. These new techniques are compared to six well-known cell formation algorithms in terms of different efficiency measures according to randomly generated problems. The results are analyzed statistically.Kandiller, LeventPh.D