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Lower Bounds on Flow Sparsifiers with Steiner Nodes

Yu Chen, Zihan Tan, Mingyang Yang

TL;DR

There exist $k-terminal graphs such that, even if the authors allow $k\cdot 2^{(\log k)^{\Omega(1)}}$ Steiner nodes in its contraction-based flow sparsifier, the quality is still $\Omega\big((\log k)^{0.3}\big)$.

Abstract

Given a large graph $G$ with a set of its $k$ vertices called terminals, a \emph{quality-$q$ flow sparsifier} is a small graph $G'$ that contains the terminals and preserves all multicommodity flows between them up to some multiplicative factor $q\ge 1$, called the \emph{quality}. Constructing flow sparsifiers with good quality and small size ($|V(G')|$) has been a central problem in graph compression. The most common approach of constructing flow sparsifiers is contraction: first compute a partition of the vertices in $V(G)$, and then contract each part into a supernode to obtain $G'$. When $G'$ is only allowed to contain all terminals, the best quality is shown to be $O(\log k/\log\log k)$ and $Ω(\sqrt{\log k/\log\log k})$. In this paper, we show that allowing a few Steiner nodes does not help much in improving the quality. Specifically, there exist $k$-terminal graphs such that, even if we allow $k\cdot 2^{(\log k)^{Ω(1)}}$ Steiner nodes in its contraction-based flow sparsifier, the quality is still $Ω\big((\log k)^{0.3}\big)$.

Lower Bounds on Flow Sparsifiers with Steiner Nodes

TL;DR

There exist k\cdot 2^{(\log k)^{\Omega(1)}}\Omega\big((\log k)^{0.3}\big)$.

Abstract

Given a large graph with a set of its vertices called terminals, a \emph{quality- flow sparsifier} is a small graph that contains the terminals and preserves all multicommodity flows between them up to some multiplicative factor , called the \emph{quality}. Constructing flow sparsifiers with good quality and small size () has been a central problem in graph compression. The most common approach of constructing flow sparsifiers is contraction: first compute a partition of the vertices in , and then contract each part into a supernode to obtain . When is only allowed to contain all terminals, the best quality is shown to be and . In this paper, we show that allowing a few Steiner nodes does not help much in improving the quality. Specifically, there exist -terminal graphs such that, even if we allow Steiner nodes in its contraction-based flow sparsifier, the quality is still .
Paper Structure (37 sections, 20 theorems, 36 equations, 3 figures)

This paper contains 37 sections, 20 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Technical Overview
  4. Paths in $H$ and paths in $G$.
  5. Necessity of Requirement 2 ($k\ge n/2^{(\log n)^{\alpha}}$).
  6. Related Work
  7. Other recent results on flow sparsifiers.
  8. Cut sparsifiers.
  9. Preliminaries
  10. Partitions and contracted graphs.
  11. Demands, flows, and congestions.
  12. Flow sparsifiers and their quality.
  13. Proof of \ref{['thm: main']} for Contraction-based Flow sparsifiers
  14. Expanders from random matchings
  15. Construction of the hard instance: graph and edge capacities
  16. ...and 22 more sections

Key Result

Theorem 0

For every large enough integer $k$ and every small constant ${\varepsilon}>0$, there exists a graph $G$ with $k$ terminals, such that any convex combination of contraction-based flow sparsifiers on at most $k\cdot 2^{(\log k)^{(1-c_0)+{\varepsilon}}}$ vertices has quality at least $(\log k)^{c_0-O({

Figures (3)

  • Figure 1: An explanation of why the diameter increases by $5$ times per level up: A level-$(i+1)$ cluster consists of five level-$i$ clusters, with the red one being the center, the green ones its distance-$1$ neighbors, and the black ones distance-$2$ clusters assigned to it. The blue dashed line shows the shortest path between a pair of vertices in this level-$(i+1)$ cluster, whose length is at most $5\cdot \textnormal{diam}(i)+4$.
  • Figure 2: An illustration of increasing edge capacities.
  • Figure 3: Proper partition result $\mathcal{A}=(C,\mathcal{H})=(C^0,\ldots C^{l-1}, \mathcal{H}^1,\ldots, \mathcal{H}^{l})$.

Theorems & Definitions (38)

  • Theorem 0
  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2: puder2015expansion
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 4
  • ...and 28 more