Tight Bounds for Gomory-Hu-like Cut Counting

By a classical result of Gomory and Hu (1961), in every edgeweighted graph G = (V,E,w), the minimum st-cut values, when ranging over all s, t ∈ V , take at most |V |−1 distinct values. That is, these (|V | 2 ) instances exhibit redundancy factor Ω(|V |). They further showed how to construct from G a tree (V,E′, w′) that stores all minimum st-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum st-cut problem. 1. Group-Cut: Consider the minimum (A,B)-cut, ranging over all subsets A,B ⊆ V of given sizes |A| = α and |B| = β. The redundancy factor is Ωα,β(|V |). 2. Multiway-Cut: Consider the minimum cut separating every two vertices of S ⊆ V , ranging over all subsets of a given size |S| = k. The redundancy factor is Ωk(|V |). 3. Multicut: Consider the minimum cut separating every demand-pair in D ⊆ V × V , ranging over collections of |D| = k demand pairs. The redundancy factor is Ωk(|V |). This result is a bit surprising, as the redundancy factor is much larger than in the first two problems. A natural application of these bounds is to construct small data structures that stores all relevant cut values, à la the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.