Multicut Survey

Graph cut problems are widely studied in the area of approximation algorithms. The most basic cut problem is the s-t minimum cut problem, for which Ford and Fulkerson gave an exact algorithm and illustrated max-flow min-cut relationship. This min-max theorem has led researchers to seek its generalization to the case of multicommodity flow. In this setting, each commodity has its own source and sink, and the object is to maximaze the sum of the flows subject to capacity and flow conservation requirements. The notion of a multicut generalizes that of a cut, and is defined as a set of edges whose removal disconnects each source from its corresponding sink. Clearly, maximum multicommodity flow is bounded by minimum multicut; however, the equality doesn’t hold often times for k ≥ 3, where k is the number of source and sink pairs. And minimum multicut is proved to be NP-hard when k ≥ 3. In this situation, the best one can hope for is an approximation algorithm for minimum multicut problems. This suvey collects approximation and complexity results for multicut problems and its variants.