Vertex packings: Structural properties and algorithms

We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.

[1]  D. R. Fulkerson,et al.  Anti-blocking polyhedra , 1972 .

[2]  Arthur M. Geoffrion,et al.  An Improved Implicit Enumeration Approach for Integer Programming , 1969, Oper. Res..

[3]  Claude Berge,et al.  The theory of graphs and its applications , 1962 .

[4]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..

[5]  Leslie E. Trotter,et al.  Properties of vertex packing and independence system polyhedra , 1974, Math. Program..

[6]  J. Trotter.  Solution characteristics and algorithms for the vertex packing problem. , 1973 .

[7]  Kurt Spielberg,et al.  Set Covering by Single-Branch Enumeration with Linear-Programming Subproblems , 1971, Oper. Res..

[8]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[9]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[10]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[11]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[12]  Egon Balas,et al.  Finding a Minimum Node Cover in an Arbitrary Graph. , 1973 .

[13]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[14]  Leslie E. Trotter,et al.  A class of facet producing graphs for vertex packing polyhedra , 1975, Discret. Math..

[15]  R. Tarjan.  Finding a Maximum Clique. , 1972 .

[16]  J. Edmonds.  Covers and packings in a family of sets , 1962 .