Tight Bounds for Gomory-Hu-like Cut Counting


By a classical result of Gomory and Hu (1961), in every edge-weighted graph G=(V,E,w)G=(V,E,w), the minimum stst-cut values, when ranging over all s,tVs,t\in V, take at most V1|V|-1 distinct values. That is, these (V2)\binom{|V|}{2} instances exhibit redundancy factor Ω(V)\Omega(|V|). They further showed how to construct from GG a tree (V,E,w)(V,E',w') that stores all minimum stst-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum stst-cut problem. 1. Group-Cut: Consider the minimum (A,B)(A,B)-cut, ranging over all subsets A,BVA,B\subseteq V of given sizes A=α|A|=\alpha and B=β|B|=\beta. The redundancy factor is Ωα,β(V)\Omega_{\alpha,\beta}(|V|). 2. Multiway-Cut: Consider the minimum cut separating every two vertices of SVS\subseteq V, ranging over all subsets of a given size S=k|S|=k. The redundancy factor is Ωk(V)\Omega_{k}(|V|). 3. Multicut: Consider the minimum cut separating every demand-pair in DV×VD\subseteq V\times V, ranging over collections of D=k|D|=k demand pairs. The redundancy factor is Ωk(Vk)\Omega_{k}(|V|^k). 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, like the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which have some overlap with our results), see Bibliographic Update 1.

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