Threshold methods for boolean optimization problems with separable objectives

Abstract

Let P denote the set of all subsets of N:={1,2,...,n}. Let (H,*,≤) be a negatively ordered commutative semigroup with internal composition "*" and order relation "≤". Separable objectives f : P → H have the general form $f(x) = \mathop *\limits_{i \in x} c_i$ with coefficients c_i ε H for i ε N. The separable objective shall be maximized over B_k ⊆ P which consists only of sets of cardinality k ε N. Especially the set of all intersections of maximal cardinality for two matroids is considered. From the threshold method of Edmonds and Fulkerson for bottleneck extrema we derive a class of suboptimal algorithms for the general problem. During the algorithm lower and upper bounds for the optimal objective value are determined.