Faster and simpler approximation algorithms for mixed packing and covering problems

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Abstract

We propose an algorithm for approximately solving the mixed packing and covering problem; given a convex compact set 0{combining long solidus overlay} ≠ B ⊆ RN, either compute x ∈ B such that f (x) ≤ (1 + ε{lunate}) a and g (x) ≥ (1 - ε{lunate}) b or decide that {x ∈ B {divides} f (x) ≤ a, g (x) ≥ b} = 0{combining long solidus overlay}. Here f, g : B → R+M are vectors whose components are M non-negative convex and concave functions, respectively, and a, b ∈ R+ +M are constant positive vectors. Our algorithm requires an efficient feasibility oracle or block solver which, given vectors c, d ∈ R+M and α ∈ R+, computes over(x, ̂) ∈ B such that cT f (over(x, ̂)) - dT g (over(x, ̂)) ≤ α or correctly decides that no such over(x, ̂) ∈ 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 + ε{lunate}- 2 ln ε{lunate}- 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. © 2007 Elsevier Ltd. All rights reserved.

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Diedrich, F., & Jansen, K. (2007). Faster and simpler approximation algorithms for mixed packing and covering problems. Theoretical Computer Science, 377(1–3), 181–204. https://doi.org/10.1016/j.tcs.2007.02.064

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