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

Pierre Bergé, Benjamin Mouscadet, Arpad Rimmel, Joanna Tomasik

Abstract: 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 $$n=\left| V \right| $$ 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 $$2^{O(p^2)}n^{O(1)}$$ . Initially dedicated to counting minimum cuts, it can be modified to obtain an FPT sampling of minimum edge (S, T)-cuts.