An Approximation Algorithm for the General Mixed Packing and Covering Problem

Abstract

We present a price-directive decomposition algorithm to compute an approximate solution of the mixed packing and covering problem; it either finds x ∈ B such that f(x) ≤ c(1 + ε)a and g(x) ≥ (1 − ε)b/c or correctly decides that {x ∈ B|f(x) ≤ a, g(x) ≥ b} = ∅. Here f,g are vectors of M ≥ 2 convex and concave functions, respectively, which are nonnegative on the convex compact set ∅ ≠ B ⊆ ℝ^N; B can be queried by a feasibility oracle or block solver, a, $b\in \mathbb{R}^{M}_{++}$ and c is the block solver’s approximation ratio. The algorithm needs only O(M(ln M + ε ^− 2 ln ε ^− 1)) iterations, a runtime bound independent from c and the input data. Our algorithm is a generalization of [16] and also approximately solves the fractional packing and covering problem where f,g are linear and B is a polytope; there, a width-independent runtime bound is obtained.

Research supported in part by DFG Project “Entwicklung und Analyse von Approximativen Algorithmen für Gemischte und Verallgemeinerte Packungs- und Überdeckungsprobleme” JA 612/10-1, in part by EU Project AEOLUS IST-15964, and in part by a PPP funding “Scheduling in Communication Networks” D/05/06936 of the DAAD.