Unbalanced Graph Cuts

We introduce the Minimum-size bounded-capacity cut (MinSBCC) problem, in which we are given a graph with an identified source and seek to find a cut minimizing the number of nodes on the source side, subject to the constraint that its capacity not exceed a prescribed bound B. Besides being of interest in the study of graph cuts, this problem arises in many practical settings, such as in epidemiology, disaster control, military containment, as well as finding dense subgraphs and communities in graphs. In general, the MinSBCC problem is NP-complete. We present an efficient $(\frac{1}{{\rm \lambda}},\frac{1}{1-{\rm \lambda}})$-bicriteria approximation algorithm for any 0 < λ < 1; that is, the algorithm finds a cut of capacity at most $\frac{1}{{\rm \lambda}}B$, leaving at most $\frac{1}{1-{\rm \lambda}}$ times more vertices on the source side than the optimal solution with capacity B. In fact, the algorithm’s solution either violates the budget constraint, or exceeds the optimal number of source-side nodes, but not both. For graphs of bounded treewidth, we show that the problem with unit weight nodes can be solved optimally in polynomial time, and when the nodes have weights, approximated arbitrarily well by a PTAS.