Tight Bounds for Gomory-Hu-like Cut Counting

TL;DR

This paper establishes tight bounds on the redundancy factors for various generalizations of the minimum cut problem, extending the classical Gomory-Hu result to more complex cut scenarios.

Contribution

It provides the first tight bounds on redundancy factors for group-cut, multiway-cut, and multicut problems, generalizing Gomory-Hu's classical theorem.

Findings

Redundancy factor for group-cut is _{\u03b1,}(|V|).

Redundancy factor for multiway-cut is _{k}(|V|).

Redundancy factor for multicut is _{k}(|V|^k).

Abstract

By a classical result of Gomory and Hu (1961), in every edge-weighted graph $G=(V,E,w)$, the minimum $st$-cut values, when ranging over all $s,t\in V$, take at most $|V|-1$ distinct values. That is, these $\binom{|V|}{2}$ instances exhibit redundancy factor $\Omega(|V|)$. 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 $A,B\subseteq V$ of given sizes $|A|=\alpha$ and $|B|=\beta$. The redundancy factor is $\Omega_{\alpha,\beta}(|V|)$. 2. Multiway-Cut: Consider the minimum cut separating every two vertices of $S\subseteq V$, ranging over all subsets of a given size $|S|=k$. The redundancy factor is $\Omega_{k}(|V|)$. 3. Multicut: Consider the minimum cut separating every demand-pair in $D\subseteq V\times V$, ranging over collections of $|D|=k$ demand pairs. The redundancy factor is $\Omega_{k}(|V|^k)$. 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.