abstract
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Many load balancing problems that arise in scientific computing applications boil down to the problem of partitioning a graph with weights on the vertices and costs on the edges into a given number of equally-weighted parts such that the maximum boundary cost over all parts is small.
Here, this partitioning problem is considered for graphs \(G=(V,E)\) with edge costs \(c\from E\to \Rnn\), that have bounded maximum degree and 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 separator \(S\subseteq V\) such that the edges incident to \(S\) in the subgraph have total cost at most proportional to \((\sum_e c^p_e)^{1/p}\), where the sum is over all edges in the subgraph.
For arbitrary weights \(w\from V\to \Rnn\), we show that the vertices of such graphs can be partitioned into \(k\) parts such that the weight of each part differs from the average weight \(\sum_{v\in V}w_v/k\) by at most \((1-\frac{1}{k})\max_{v\in V}w_v\), and the boundary edges of each part have total cost at most proportional to \((\sum_{e\in E}c_e^p/k)^{1/p}+\max_{e\in E}c_e\). The partition can be computed in time nearly proportional to the time for computing separators \(S\) for \(G\) as above.
Our upper bound 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 \(c\equiv 1\), \(w\equiv 1\), and one allows parts of weight as large as a constant multiple of the average weight.
We also give a separator theorem for \(d\)-dimensional grid graphs with arbitrary edge costs, which is the first result of its kind for non-planar graphs.