Tight Bounds for GomoryHulike Cut Counting
Abstract
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 $s,t\in V$, take at most $V1$ 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. GroupCut: 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. MultiwayCut: 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 demandpair 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 GomoryHu tree. We initiate this direction by giving some upper and lower bounds.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.08647
 Bibcode:
 2015arXiv151108647C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics
 EPrint:
 This version contains additional references to previous work (which have some overlap with our results), see Bibliographic Update 1.1