Approximating Minimum Cut with Bounded Size

Abstract

We present the Minimum Cut with Bounded Size problem and two efficient algorithms for its solution. In this problem we want to partition the n vertices of a edge-weighted graph into two sets S and T, with S including a given source s, T a given sink t, and with |S| bounded by a given threshold B, so as to minimize the weight δ(S) of the edges crossing the cut (S,T). If B is equal to n − 1 the problem is well-known to be solvable in polynomial time, but for general B it becomes NP-hard. The first algorithm is randomized and, for each ε > 0, it returns, with high probability, a solution S having a weight within ratio $(1+\frac{\varepsilon B}{\log n})$ of the optimum. The second algorithm is a deterministic bi-criteria algorithm which can return a solution violating the cardinality constraint within a specified ratio; precisely, for each 0 < γ < 1, it returns a set S having either (1) a weight within ratio $\frac{1}{1-\gamma }$ of the optimum or (2) optimum weight but cardinality $\ |S|\leq \frac{B}{\gamma }$, and hence it violates the constraint by a factor at most $\frac{1}{\gamma}$.