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Towards the Characterization of Terminal Cut Functions: a Condition for Laminar Families

Yu Chen, Zihan Tan

Abstract

We study the following characterization problem. Given a set $T$ of terminals and a $(2^{|T|}-2)$-dimensional vector $π$ whose coordinates are indexed by proper subsets of $T$, is there a graph $G$ that contains $T$, such that for all subsets $\emptyset\subsetneq S\subsetneq T$, $π_S$ equals the value of the min-cut in $G$ separating $S$ from $T\setminus S$? The only known necessary conditions are submodularity and a special class of linear inequalities given by Chaudhuri, Subrahmanyam, Wagner and Zaroliagis. Our main result is a new class of linear inequalities concerning laminar families, that generalize all previous ones. Using our new class of inequalities, we can generalize Karger's approximate min-cut counting result to graphs with terminals.

Towards the Characterization of Terminal Cut Functions: a Condition for Laminar Families

Abstract

We study the following characterization problem. Given a set of terminals and a -dimensional vector whose coordinates are indexed by proper subsets of , is there a graph that contains , such that for all subsets , equals the value of the min-cut in separating from ? The only known necessary conditions are submodularity and a special class of linear inequalities given by Chaudhuri, Subrahmanyam, Wagner and Zaroliagis. Our main result is a new class of linear inequalities concerning laminar families, that generalize all previous ones. Using our new class of inequalities, we can generalize Karger's approximate min-cut counting result to graphs with terminals.
Paper Structure (8 sections, 3 theorems, 23 equations, 2 figures)

This paper contains 8 sections, 3 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work.
  4. Organization.
  5. Proof of \ref{['lem:cover']}
  6. Proof of \ref{['thm:terminal-karger']}
  7. Discussion and Future Work
  8. Acknowledgement.

Key Result

Theorem 1

Let $G$ be an edge-capacitated graph, $T$ be a set of its vertices, and $\beta,\gamma$ be type vectors. If (i) $\mathsf{supp}(\gamma)$ is a laminar family; and (ii) metric $\mathsf{D}_{\beta}$ dominates metric $\mathsf{D}_{\gamma}$ (that is, for every pair $t,t'\in T$, $\mathsf{D}_{\beta}(t,t') \ge

Figures (2)

  • Figure 1: An illustration of the continuization of a graph. Line $\textnormal{con}(u,v)$ has length $3$ and a point $p$ in this line is $p=(u,1)=(v,2)$. Line $\textnormal{con}(u',v')$ has length $3$ and a point $p'$ in this line is $p'=(u',2)=(v',1)$. The shortest (in $\ell^{\textnormal{con}}$) path connecting $p$ to $p'$ in $V^{\textnormal{con}}$ is shown in the red dashed line.
  • Figure 2: An illustration of balls and regions. The terminals in $S$ are shown in red. The balls $\left\{ \mathsf{B}(t,r_t) \right\}_{t\in S}$ are shown in green. The region $\Phi_S$ is the union of all sub-segments shown in purple.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • Claim 5
  • proof
  • Claim 6
  • proof
  • Claim 7
  • proof
  • ...and 4 more