Facets of conflict hypergraphs

Facet;Valid inequality;Conflict hypergraph;Conflict graph;Independent set polytope;Independent set problem;Combinatorial optimization;Integer programming;Market split;Market share;Parallel computing

Maheshwary, Siddhartha ; Industrial and Systems Engineering

University:Georgia Institute of Technology

Department:Industrial and Systems Engineering

关键词: Facet;    Valid inequality;    Conflict hypergraph;    Conflict graph;    Independent set polytope;    Independent set problem;    Combinatorial optimization;    Integer programming;    Market split;    Market share;    Parallel computing;   

Others  :  https://smartech.gatech.edu/bitstream/1853/26635/1/maheshwary_siddhartha_200812_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

We study the facial structure of the independent set polytope using the concept of conflict hypergraphs. A conflict hypergraph is a hypergraph whose vertices correspond to the binary variables, and edges correspond to covers in the constraint matrix of the independent set polytope. Variousstructures such as cliques, odd holes, odd anti-holes, webs and anti-webs are identified on the conflict hypergraph. These hypergraph structures are shown to be generalization of traditional graph structures.Valid inequalities are derived from these hypergraph structures, and the facet defining conditions are studied. Chvatal-Gomory ranks are derived for odd hole and clique inequalities. To test the hypergraph cuts, we conduct computational experiments on market-share (also referred to as market-split) problems. These instances consist of 100% dense multiple-knapsack constraints. They are small in size but are extremely hard to solve by traditional means. Their difficult nature is attributed mainly to the dense and symmetrical structure. We employ a special branching strategy in combination with the hypergraph inequalities to solve many of the particularly difficult instances. Results are reported for serial as well as parallel implementations.