## Roof duality, complementation and persistency in quadratic 0-1 optimization.(English)Zbl 0574.90066

The paper is concerned with the ”primal” problem of maximizing a given quadratic pseudoboolean 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 majorants p(x) of f, having the following persistency property: if the ith coefficient in p is positive (negative) then $$x_ i=1(0)$$ 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.

### MSC:

 90C09 Boolean programming 90C30 Nonlinear programming 65K05 Numerical mathematical programming methods
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### References:

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