Convex Combinatorial Optimization

We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications

Direct Product Decompositions of Lattices, Closures and Relation Schemes

In this paper we study direct product decompositions of closure operations and lattices of closed sets. We characterize direct product decompositions of lattices of closed sets in terms of closure operations, and find those decompositions of lattices which correspond to the decompositions of closures. If a closure on a finite set is represented by its implication base (i.e. a binary relation on a powerset), we construct a polynomial algorithm to find its direct product decompositions. The main characterization theorem is also applied to define direct product decompositions of relational database schemes and to find out what properties of relational databases and schemes are preserved under decompositions

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