Approximating Polymatroid Packing and Covering

Toshihiro FUJITO  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E85-A   No.5   pp.1066-1070
Publication Date: 2002/05/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section LETTER (Special Section on Discrete Mathematics and Its Applications)
combinatorial problems,  design of algorithms,  optimization,  

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We consider the polymatroid packing and covering problems. The polynomial time algorithm with the best approximation bound known for either problem is the greedy algorithm, yielding guaranteed approximation factors of 1/k for polymatroid packing and H(k) for polymatroid covering, where k is the largest rank of an element in a polymatroid, and H(k)=Σi=1k 1/i is the kth Harmonic number. The main contribution of this note is to improve these bounds by slightly extending the greedy heuristics. Specifically, it will be shown how to obtain approximation factors of 2/(k+1) for packing and H(k)-1/6 for covering, generalizing some existing results on k-set packing, matroid matching, and k-set cover problems.