Coordination complexity of parallel price-directive decomposition
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STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
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Theoretical Computer Science
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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} ≤ θ. Here f, g are vectors of M ≤ 2 convex and concave functions, respectively, which are nonnegative on the convex compact set θ ≠ ≤ B ⊆ RN; B can be queried by a feasibility oracle or block solver, a, b ∈ RM++ 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.