# Tight bounds for Gomory-Hu-like cut counting

Rajesh Chitnis*, Lior Kamma, Robert Krauthgamer

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

## Abstract

By a classical result of Gomory and Hu (1961), in every edge-weighted graph G = (V, E, ω), 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, Eʹ, ωʹ) 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|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, à la the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.

Original language English Graph-Theoretic Concepts in Computer Science 42nd International Workshop, WG 2016, Istanbul, Turkey, June 22-24, 2016, Revised Selected Papers Pinar Heggernes Springer Verlag 133-144 9783662535363 9783662535356 https://doi.org/10.1007/978-3-662-53536-3_12 Published - 28 Sept 2016 42nd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2016 - Istanbul, TurkeyDuration: 22 Jun 2016 → 24 Jun 2016

### Publication series

Name Lecture Notes in Computer Science Springer 9941 0302-9743 1611-3349

### Conference

Conference 42nd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2016 Turkey Istanbul 22/06/16 → 24/06/16

## ASJC Scopus subject areas

• Theoretical Computer Science
• General Computer Science

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