2006/06/01 by David Steurer, Steurer, David
Computer Science · #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #G.2.2 #cs.DM #cs.DS
paper · pdf · doi:10.48550/arxiv.cs/0606001
41 pages, full version of a paper that will appear in SPAA`06
arxiv created 2006/06/01 · arxiv updated 2009/12/01
Many load balancing problems that arise in scientific computing applications ask to partition a graph with weights on the vertices and costs on the edges into a given number of almost equally-weighted parts such that the maximum boundary cost over all parts is small. Here, this partitioning problem is considered for bounded-degree graphs G=(V,E) with edge costs c: E->R+ that have a p-separator theorem for some p>1, i.e., any (arbitrarily weighted) subgraph of G can be separated into two parts of roughly the same weight by removing a vertex set S such that the edges incident to S in the subgraph have total cost at most proportional to (SUMe cpe)^(1/p), where the sum is over all edges e in the subgraph. We show for all positive integers k and weights w that the vertices of G can be partitioned into k parts such that the weight of each part differs from the average weight by less than MAXwv; v in V, and the boundary edges of each part have cost at most proportional to (SUMe cep/k)^(1/p) + MAXe ce. The partition can be computed in time nearly proportional to the time for computing a separator S of G. Our upper bound on the boundary costs is shown to be tight up to a constant factor for infinitely many instances with a broad range of parameters. Previous results achieved this bound only if one has c=1, w=1, and one allows parts with weight exceeding the average by a constant fraction.