Partitioning Trees with Supply, Demand and Edge-Capacity


IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E96-A   No.6   pp.1036-1043
Publication Date: 2013/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E96.A.1036
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
tree,  maximum partition problem,  supply,  demand,  edge-capacity,  approximation algorithm,  

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Let T be a given tree. Each vertex of T is either a supply vertex or a demand vertex, and is assigned a positive number, called the supply or demand. Each demand vertex v must be supplied an amount of “power,” equal to the demand of v, from exactly one supply vertex through edges in T. Each edge is assigned a positive number called the capacity. One wishes to partition T into subtrees by deleting edges from T so that each subtree contains exactly one supply vertex whose supply is no less than the sum of all demands in the subtree and the power flow through each edge is no more than capacity of the edge. The “partition problem” is a decision problem to ask whether T has such a partition. The “maximum partition problem” is an optimization version of the partition problem. In this paper, we give three algorithms for the problems. First is a linear-time algorithm for the partition problem. Second is a pseudo-polynomial-time algorithm for the maximum partition problem. Third is a fully polynomial-time approximation scheme (FPTAS) for the maximum partition problem.