Lecture notes for “ Analysis of Algorithms ” : Global minimum cuts ( Draft )

We describe deterministic and randomized algorithms for finding global minimum cuts in undirected graphs. 1 Global and s-t cuts in undirected graphs Let G = (V,E,w) be a weighted undirected graph, where w : E → R is a weight (or cost, or capacity) function defined on its edges. A (global) cut {S, T} of G is a partition of V into two disjoint non-empty sets, i.e., S, T 6= φ and S ∩ T = φ. Edges in the set E(S, T ) = {e ∈ E | |e ∩ S| = 1} are said to cross the cut. Let s, t ∈ V . A cut {S, T} is said to be an s-t cut if and only if |{s, t}∩S| = 1, i.e., s ∈ S and t ∈ T , or s ∈ T and t ∈ S. The weight w(S, T ) of a cut {S, T} is defined to be the sum of the weights of the edges that cross the cut: