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Improved Lower Bounds on Multiflow-Multicut Gaps

Sina Kalantarzadeh, Nikhil Kumar

TL;DR

This work significantly improves our understanding of the multiflow-multicut gap in planar settings by establishing a new lower bound of $\frac{16}{7}$ for cactus graphs, thus surpassing the long-standing bound of 2. The authors introduce a general, nonconstructive framework that connects the integrality gap of the minimum multicut LP to small-diameter decompositions (SDD), enabling a transfer of tree-based techniques to broader graph families via 1-sums and minor-closed classes. They first derive a $\tfrac{20}{9}$ lower bound, then refine it to $\tfrac{16}{7}$ for cactus graphs, and provide explicit constructions achieving the $\tfrac{20}{9}$ limit. The work combines new decomposition-based methods with carefully crafted cactus gadgets to derive both explicit and structural lower bounds, paving the way for sharper constants in planar graph subclasses and raising open questions about tight gaps in these families.

Abstract

Given a set of source-sink pairs, the maximum multiflow problem asks for the maximum total amount of flow that can be feasibly routed between them. The minimum multicut, a dual problem to multiflow, seeks the minimum-cost set of edges whose removal disconnects all the source-sink pairs. It is easy to see that the value of the minimum multicut is at least that of the maximum multiflow, and their ratio is called the multiflow-multicut gap. The classical max-flow min-cut theorem states that when there is only one source-sink pair, the gap is exactly one. However, in general, it is well known that this gap can be arbitrarily large. In this paper, we study this gap for classes of planar graphs and establish improved lower bound results. In particular, we show that this gap is at least $\frac{16}{7}$ for the class of planar graphs, improving upon the decades-old lower bound of 2. More importantly, we develop new techniques for proving such a lower bound, which may be useful in other settings as well.

Improved Lower Bounds on Multiflow-Multicut Gaps

TL;DR

This work significantly improves our understanding of the multiflow-multicut gap in planar settings by establishing a new lower bound of for cactus graphs, thus surpassing the long-standing bound of 2. The authors introduce a general, nonconstructive framework that connects the integrality gap of the minimum multicut LP to small-diameter decompositions (SDD), enabling a transfer of tree-based techniques to broader graph families via 1-sums and minor-closed classes. They first derive a lower bound, then refine it to for cactus graphs, and provide explicit constructions achieving the limit. The work combines new decomposition-based methods with carefully crafted cactus gadgets to derive both explicit and structural lower bounds, paving the way for sharper constants in planar graph subclasses and raising open questions about tight gaps in these families.

Abstract

Given a set of source-sink pairs, the maximum multiflow problem asks for the maximum total amount of flow that can be feasibly routed between them. The minimum multicut, a dual problem to multiflow, seeks the minimum-cost set of edges whose removal disconnects all the source-sink pairs. It is easy to see that the value of the minimum multicut is at least that of the maximum multiflow, and their ratio is called the multiflow-multicut gap. The classical max-flow min-cut theorem states that when there is only one source-sink pair, the gap is exactly one. However, in general, it is well known that this gap can be arbitrarily large. In this paper, we study this gap for classes of planar graphs and establish improved lower bound results. In particular, we show that this gap is at least for the class of planar graphs, improving upon the decades-old lower bound of 2. More importantly, we develop new techniques for proving such a lower bound, which may be useful in other settings as well.

Paper Structure

This paper contains 12 sections, 13 theorems, 107 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Related Work: Demand Multicommodity Flow
  3. Preliminaries
  4. Linear Programming Relaxation for the Minimum Multicut Problem
  5. Our Results and Techniques
  6. The Integrality Gap of Multicut and Small Diameter Decomposition
  7. Small Diameter Decomposition for Trees
  8. A Structural Result for Small Diameter Decompositions
  9. $\frac{20}{9}$ Lower Bound For Cactus Graphs
  10. $\tfrac{16}{7}$ Lower Bound for Cactus Graphs
  11. An Explicit Construction for the $\frac{20}{9}$ Lower Bound
  12. Conclusions and Future Work

Key Result

Theorem 1

Suppose we are given an instance $M \in \mathcal{M}(\mathcal{G})$. Let $\mathcal{F} \subseteq 2^{E(G)}$ be the set of all feasible multicuts for $M$, and let $x$ be a feasible fractional solution to the LP relaxation LP of multicut. Then there exists a probability distribution $y$ over $\mathcal{F}$

Figures (11)

  • Figure 1: $G$ is a simple path with $3$ edges, and $k=2$. Let $F_{1}=\{e_{1},e_{2}\},F_{2}=\{e_{2}\},F_{3}=\{e_{2},e_{3}\},F_{4}=\{e_{1},e_{3}\}$. One can see that $\mathcal{F}_{2}(G)=\{F_{1},F_{2},F_{3},F_{4},E(G)\}$. Let $y_{F_{1}}=y_{F_{3}}=y_{E(G)}=0$, and $y_{F_{2}}=y_{F_{4}}=\frac{1}{2}$. This distribution is a $SDD(G,2,\frac{1}{2})$.
  • Figure 2: An illustration of the 1-sum operation
  • Figure 3: Illustration of the construction of $H$
  • Figure 4: The construction of $G^{S(L)}$ with $L=\{(u_{1},(G_{1},r_{1})),(u_{2},(G_{2},r_{2})),(u_{3},(G_{3},r_{3}))\}$.
  • Figure 5: The left graph is $G^{S(L)}$ and the right graph is $G^{S(L')}$.
  • ...and 6 more figures

Theorems & Definitions (44)

  • Definition 1
  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof
  • Claim 1
  • proof : Proof of Claim
  • Theorem 3
  • proof
  • ...and 34 more