An Approximate Max-Flow Min-Cut Relation for Unidirected Multicommodity Flow, with Applications

In this paper , we prove the first approximate max-flow min-cut theorem for undirected mul t i commodi ty flow. We show tha t for a feasible flow to exist in a mul t icommodi ty problem, it is sufficient t ha t every cut ' s capacity exceeds its demand by a factor of O(logClogD), where C is the sum of all finite capacities and D is the sum of demands . Moreover, our theorem yields an a lgor i thm for finding a cut tha t is approximately min imum relative to the flow tha t must cross it. We use this result to obta in an approximat ion algori thm for T. C. Hu's generalization of the mult iway-cut problem. This a lgori thm can in turn be applied to obtain approximat ion algori thms for min imum delet ion of clauses of a 2-CNF-formula, via minimization, and other problems. We also generalize the theorem to hypergraph networks; using this generalization, we can handle CNFclauses wi th an arbi t rary number of literals per clause.