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Journal of the ACM (JACM)
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Information Processing Letters
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Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
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Journal of Algorithms
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STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Approximating the Domatic Number
SIAM Journal on Computing
Random Structures & Algorithms - Probabilistic methods in combinatorial optimization
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
A new multilayered PCP and the hardness of hypergraph vertex cover
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Packing element-disjoint steiner trees
ACM Transactions on Algorithms (TALG)
A Graph Reduction Step Preserving Element-Connectivity and Applications
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Cover-decomposition and polychromatic numbers
ESA'11 Proceedings of the 19th European conference on Algorithms
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Let f : 2N → $\cal {Z}$+ be a polymatroid (an integer-valued non-decreasing submodular set function with f(∅) = 0). We call S ⊆ N a base if f(S) = f(N). We consider the problem of finding a maximum number of disjoint bases; we denote by m* be this base packing number. A simple upper bound on m* is given by k* = max{k : ΣiεNfA(i) ≥ kfA(N),∀A ⊆ N} where fA(S) = f(A ∪ S) - f(A). This upper bound is a natural generalization of the bound for matroids where it is known that m* = k*. For polymatroids, we prove that m* ≥ (1 - o(1))k*-lnf(N) and give a randomized polynomial time algorithm to find (1 - o(1))k*-lnf(N) disjoint bases, assuming an oracle for f. We also derandomize the algorithm using minwise independent permutations and give a deterministic algorithm that finds (1 - ε)k*-lnf(N) disjoint bases. The bound we obtain is almost tight because it is known there are polymatroids for which m* o(1))k*-lnf(N). Moreover it is known that unless NP ⊆ DTIME(nlog log n), for any ε 0, there is no polynomial time algorithm to obtain a (1 + ε)-lnf(N)-approximation to m*. Our result generalizes and unifies two results in the literature. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009