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Improved Upper Bounds for the Directed Flow-Cut Gap

Greg Bodwin, Luba Samborska

Abstract

We prove that the flow-cut gap for $n$-node directed graphs is at most $n^{1/3 + o(1)}$. This is the first improvement since a previous upper bound of $\widetilde{O}(n^{11/23})$ by Agarwal, Alon, and Charikar (STOC '07), and it narrows the gap to the current lower bound of $\widetildeΩ(n^{1/7})$ by Chuzhoy and Khanna (JACM '09). We also show an upper bound on the directed flow-cut gap of $W^{1/2}n^{o(1)}$, where $W$ is the sum of the minimum fractional cut weights. As an auxiliary contribution, we significantly expand the network of reductions among various versions of the directed flow-cut gap problem. In particular, we prove near-equivalence between the edge and vertex directed flow-cut gaps, and we show that when parametrizing by $W$, one can assume unit capacities and uniform fractional cut weights without loss of generality.

Improved Upper Bounds for the Directed Flow-Cut Gap

Abstract

We prove that the flow-cut gap for -node directed graphs is at most . This is the first improvement since a previous upper bound of by Agarwal, Alon, and Charikar (STOC '07), and it narrows the gap to the current lower bound of by Chuzhoy and Khanna (JACM '09). We also show an upper bound on the directed flow-cut gap of , where is the sum of the minimum fractional cut weights. As an auxiliary contribution, we significantly expand the network of reductions among various versions of the directed flow-cut gap problem. In particular, we prove near-equivalence between the edge and vertex directed flow-cut gaps, and we show that when parametrizing by , one can assume unit capacities and uniform fractional cut weights without loss of generality.

Paper Structure

This paper contains 38 sections, 31 theorems, 81 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Parametrizing the Flow-Cut Gap by Total Weight
  3. Relationship to Nearby Problems
  4. Multicut Approximation Algorithms.
  5. The Sparsest Cut Gap.
  6. (Weak) Low-Diameter Decompositions.
  7. Organization
  8. Proof of Main Results
  9. Setup, Notation, and Formal Statement of Main Results
  10. Statement of Edge Theorems.
  11. Statement of Vertex Theorems.
  12. Rephrasing of Uncapacitated, Uniform-Weight Vertex Setting.
  13. Intuition for the Proof of Lemma \ref{['lem:vcr']}
  14. Triangle Analysis.
  15. Close-Pair Separation.
  16. ...and 23 more sections

Key Result

Theorem 1

The flow-cut gap for $n$-node directed graphs is at most $\widehat{O}\left(n^{1/3}\right)$. This holds in both the edge- and vertex-capacitated settings. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: An instance $(G, P)$ with unit edge costs/capacities, on which the minimum fractional and integral cuts differ. A minimum integral cut has cost $2$ (left), while a minimum fractional cut has cost $\frac{3}{2}$ (middle), matching the maximum flow of value $\frac{3}{2}$ (right).
  • Figure 2: The network of reductions proved in this section. The parameter $W$ represents the total fractional cut weight, $W:=w(V)$ or $W:=w(E)$.

Theorems & Definitions (58)

  • Theorem 1: Main Result
  • Theorem 2
  • Theorem 3: Folklore; see Theorem \ref{['thm:vfgself']}
  • Theorem 4: See Section \ref{['sec:reductions']}
  • Theorem 5: See Theorems \ref{['thm:edgetovert']} and \ref{['thm:vfgleefg']}
  • Corollary 6
  • Corollary 7
  • Lemma 8
  • proof : Proof of Theorem \ref{['thm:mainweight']} ($W$-parametrized bound), assuming Lemma \ref{['lem:vcr']}
  • Lemma 9: Implicit in Gupta03
  • ...and 48 more