DC-Based Approximations on Graphs

We now return to the one basic design paradigm we skipped in our discussion of approximation algorithms for NP-hard optimization problems, namely divide-and-conquer (DC), and apply it to problems on graphs. The basic idea is as follows: Find a balanced cut of small capacity, i.e., a cut in which both sides contain roughly the same number of vertices and such that few edges cross the cut. Ignore the cut edges, recursively solve the problem on the subgraphs induced by each side of the cut, and then use those solutions to construct a solution to the original instance. The latter will only be approximate because we ignored the edges that cross the cut. Finding a balanced but of minimum capacity is itself an NP-hard optimization problem. We build our way towards constructing balanced cuts of small capacity by developing approximation algorithms for two problems related to multi-commodity flow: minimum multi-cut and sparsest cut. Those approximation algorithms will be based on LP relaxations and on efficient lowdistortion embeddings of graph metrics into metrics on real spaces, namely L1. We obtain a pseudo-approximation algorithm for balanced cut, and use it to develop an approximation algorithm for the minimum cut linear arrangement problem on graphs.