An extension of Lehman's theorem and ideal set functions

Lehmans theorem on the structure of minimally nonideal clutters is a fundamental result in polyhedral combinatorics. One approach to extending it has been to give a common generalization with the characterization of minimally imperfect clutters (Seb, 1998; Gasparyan etal., 2003). We give a new generalization of this kind, which combines two types of covering inequalities and works well with the natural definition of minors. We also show how to extend the notion of idealness to unit-increasing set functions, in a way that is compatible with minors and blocking operations.

[1]  Myriam Preissmann,et al.  Imperfect and Nonideal Clutters: A Common Approach , 2003, Comb..

[2]  Gérard Cornuéjols,et al.  Ideal clutters , 2002, Discret. Appl. Math..

[3]  Jonathan Wang,et al.  A new infinite family of minimally nonideal matrices , 2011, J. Comb. Theory A.

[4]  András Sebö,et al.  Characterizing Noninteger Polyhedra with 0-1 Constraints , 1998, IPCO.

[5]  Paul D. Seymour,et al.  On Lehman's Width-Length Characterization , 1990, Polyhedral Combinatorics.

[6]  William H. Cunningham,et al.  Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra , 1995, SIAM J. Discret. Math..

[7]  Manfred W. Padberg,et al.  Perfect zero–one matrices , 1974, Math. Program..

[8]  Li Feng,et al.  The complexity of recognizing linear systems with certain integrality properties , 2008, Math. Program..

[9]  F. Bruce Shepherd,et al.  Applying Lehman's theorems to packing problems , 1995, Math. Program..

[10]  A. Frank.  Connections in Combinatorial Optimization , 2011 .

[11]  François Margot,et al.  A catalog of minimally nonideal matrices , 1998, Math. Methods Oper. Res..

[12]  Alfred Lehman,et al.  On the width—length inequality , 1979, Math. Program..

[13]  Gérard Cornuéjols,et al.  Lehman matrices , 2009, J. Comb. Theory, Ser. B.

[14]  L. Lovász.  A Characterization of Perfect Graphs , 1972 .

[15]  P. Seymour,et al.  The Strong Perfect Graph Theorem , 2002, math/0212070.

[16]  Alfred Lehman,et al.  The Width-Length Inequality and Degenerate Projective Planes , 1990, Polyhedral Combinatorics.

[17]  Gérard Cornuéjols,et al.  Ideal 0, 1 Matrices , 1994, J. Comb. Theory, Ser. B.