An Approximation Algorithm for the General Mixed Packing and Covering Problem

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 + )a and g(x) ≥ (1 − )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 ; B can be queried by a feasibility oracle or block solver, a, b ∈ R++ and c is the block solver’s approximation ratio. The algorithm needs only O(M(lnM + −2 ln −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.