On dicycle packings and the linear ordering polytope

Conference · Sat Dec 31 04:00:00 EST 1994

OSTI ID:36381

Penn, M; Nubov, Z

The acyclic tournaments of order n it forms the linear ordering polytope P{sub LO}{sup n}. Let D be a digraph that induces a facet-defining inequality for P{sub LO}{sup n} that is nonequivalent to a trivial inequality or to a 3-dicycle inequality. We show that for such a digraph the following holds: the value {tau} of a minimum integral dicycle cover is greater than the value {tau}* of a minimum dicycle cover. We show that {tau}* can be found by solving a linear program of polynomial size. The generalized transitive tournaments of order n forms the polytope P{sub C}{sup n} which contains P{sub LO}{sup n}. It is known that the integral extreme points of P{sub C}{sup n} coincide with those of P{sub LO}{sup n}. In this talk we present a method for obtaining a family of extreme points of P{sub C}{sup n} with values not in {l_brace}0, 1, {1/2}{r_brace}. Mirsky (1963) raised the question of characterizing {Omega}{sub n}{sup 0}, the convex hull of the non-identity permutation matrices of order n, by a set of linear constraints. Cruse (1979) solved Mirsky`s problem by presenting an implicit description of those constraints. Brualdi and Hwang (1992) have shown an explicit set of linear inequalities that characterize {Omega}{sub n}{sup 0} for n {<=} 6. Jointly with A. Borobia we restate Cruse`s characterization in terms of dicycle covers of certain graphs, and then show that Brualdi and Hwang`s result is no longer valid for n {>=} 7.

• OSTI ID: 36381
• Report Number(s): CONF-9408161--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
EFFICIENCY
MATRICES
NUMERICAL SOLUTION
VARIATIONAL METHODS