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 0@? B@?R^N, either compute x@?B such that f(x)@?(1+@e)a and g(x)>=(1-@e)b or decide that {x@?B|f(x)@?a,g(x)>=b}=0@?. 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 @a@?R"+, computes x@?@?B such that c^Tf(x@?)-d^Tg(x@?)@?@a or correctly decides that no such 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(lnM+@e^-^2ln@e^-^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.

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