On The Boolean Quadric Forest Polytope

Abstract We study the Boolean Quadric Forest Polytope, namely, the convex hull of the “extended” edge incidence-vectors of forests of a complete graph - extended by the usual linearization of the quadratic terms. Our motivation is to provide a mathematical foundation for attacking the minimum quadratic-cost forest problem via branch-and-cut methods of integer programming. We determine several families of facets of the Boolean Quadric Forest Polytope and relate them to the Boolean Quadric Polytope as well as the Forest Polylope. We give polynomial-time separation procedures for some of the families of facets.

[1]  Jon Lee,et al.  The volume of relaxed Boolean-quadric and cut polytopes , 1997, Discret. Math..

[2]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[3]  Mitsuo Gen,et al.  An effective genetic algorithm approach to the quadratic minimum spanning tree problem , 1998, Comput. Oper. Res..

[4]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[5]  Caterina De Simone,et al.  The cut polytope and the Boolean quadric polytope , 1990, Discret. Math..

[6]  A. Assad,et al.  The quadratic minimum spanning tree problem , 1992 .

[7]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[8]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[9]  Caterina De Simone.  A note on the Boolean quadric polytope , 1996, Oper. Res. Lett..

[10]  Manfred W. Padberg,et al.  The boolean quadric polytope: Some characteristics, facets and relatives , 1989, Math. Program..

[11]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[12]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.

[13]  D. West.  Introduction to Graph Theory , 1995 .

[14]  Hanif D. Sherali,et al.  A simultaneous lifting strategy for identifying new classes of facets for the Boolean quadric polytope , 1995, Oper. Res. Lett..

[15]  R. Kipp Martin,et al.  Using separation algorithms to generate mixed integer model reformulations , 1991, Oper. Res. Lett..

[16]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..