Goemans Lecture notes on the mincut problem 1 Minimum Cuts

The minimum cut problem (or mincut problem) is to find a cut of minimum cost. If all costs are 1 then the problem becomes the problem of finding a cut with as few edges as possible. Cuts are often defined in a different, not completely equivalent, way. Define a cutset to be a set of edges whose removal disconnects the graph into at least two connected components. Minimal cutsets (a minimal cutset C is a cutset such that any proper subset of C is not anymore a cutset) can be seen to correspond to cuts δ(S) for which the subgraphs induced by S and V − S are connected. Observe that only minimal cutsets can be of minimum cost (among all cutsets) and that only cuts δ(S) for which both S and V − S induce connected components can be of minimum cost (among all cuts) since the costs are assumed to be nonnegative. For this reason, the problem of finding a cutset of minimum cost is equivalent to the problem of finding a cut δ(S) of minimum cost, namely the mincut problem. From now on, we will only look at cuts δ(S) (and not cutsets). An important variant of the mincut problem is often considered. This is the problem of finding the minimum cost cut separating two given two vertices s and t. A cut δ(S) is said to separate s and t if only one of them belongs to S. We refer to this problem as the minimum (s, t)-cut problem. As seen in lecture, the minimum (s, t)-cut problem can be solved by means of network flow algorithms. Indeed it can be reduced to a max flow problem. Given a source s and a sink t of the graph G, we have seen that