Faster and simpler approximation algorithms for mixed packing and covering problems

We propose an algorithm for approximately solving the mixed packing and covering problem; given a convex compact set ∅ = B ⊆ RN , either compute x ∈ B such that f (x) ≤ (1 + )a and g(x) ≥ (1 − )b or decide that {x ∈ B | f (x) ≤ a, g(x) ≥ b} = ∅. Here f, g : B → RM + are vectors whose components are M non-negative convex and concave functions, respectively, and a, b ∈ RM ++ are constant positive vectors. Our algorithm requires an efficient feasibility oracle or block solver which, given vectors c, d ∈ RM + and α ∈ R+, computes x̂ ∈ B such that cT f (x̂) − dT g(x̂) ≤ α or correctly decides that no such x̂ ∈ B exists. Our algorithm, which is based on the Lagrangian or price-directive decomposition method, generalizes the result from [K. Jansen, Approximation algorithm for the mixed fractional packing and covering problem, in: Proceedings of 3rd IFIP Conference on Theoretical Computer Science, IFIP TCS 2004, Kluwer, 2004, pp. 223–236; SIAM Journal on Optimization 17 (2006) 331–352] and needs only O(M(ln M + −2 ln −1)) iterations or calls to the feasibility oracle. Furthermore we show that a more general block solver can be used to obtain a more general approximation within the same runtime bound. c © 2007 Elsevier B.V. All rights reserved.