Tight bounds for the Min-Max boundary decomposition cost of weighted graphs
David Steurer · 2006
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 boundeddegree 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 separator S⊆V such that the edges incident to S in the subgraph have total cost at most proportional to (Εecp e)1/p, where the sum is over all edges 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 Εv∈V wv k by less than maxv∈V wv, and the boundary edges of each part have cost at most proportional to (Εe∈ cp e/k)1/p + maxe∈E 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.