Roof duality,complementation and persistency in quadratic 0–1 optimization |
| |
Authors: | P L Hammer P Hansen B Simeone |
| |
Institution: | (1) RUTCOR, Hill Center, Rutgers University, New Brunswick, NJ, USA;(2) Institut d'Economie Scientifique et de Gestion, Lille, France;(3) Faculté Universitaire Catholique de Mons, Belgium;(4) Istituto ‘M. Picone’ per le Applicazioni del Calcolo, CNR, Rome, Italy |
| |
Abstract: | The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent
problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a
SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the
‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations
are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones.
The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal
solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx
i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results
cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the
supermodular ones) is exhibited. |
| |
Keywords: | Nonlinear 0– 1 Programming Discrete Optimization Linearization Boolean Functions Duality Graphs Stability |
本文献已被 SpringerLink 等数据库收录! |
|