Fixed-Parameter Tractability Of Counting Small Minimum (S, T)-Cuts

The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph G = (V, E) and two disjoint sets of its vertices S, T, we design a fixed-parameter tractable algorithm which counts minimum edge (S, T)-cuts parameterized by their size p. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most n = vertical bar V vertical bar successive minimum (S, T)-cuts Z(i). We prove that any minimum (S, T)-cut X contains edges of at least one cut Z(i). This observation, together with Menger's theorem, allows us to build the algorithm counting all minimum (S, T)-cuts with running time 2(O(p2)) n(O(1)). Initially dedicated to counting minimum cuts, it can be modified to obtain an FPT sampling of minimum edge (S, T)-cuts.