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Improved Approximation Bounds for Minimum Weight Cycle in the CONGEST Model

Vignesh Manoharan, Vijaya Ramachandran

TL;DR

This paper focuses on round complexity bounds for approximating MWC in the CONGEST model, and shows that for any constant $\alpha>1$, computing an $\alpha$-approximation of MWC requires $\Omega (\sqrt n}{\log n})$ rounds on weighted undirected graphs and on directed graphs, even if unweighted.

Abstract

Minimum Weight Cycle (MWC) is the problem of finding a simple cycle of minimum weight in a graph $G=(V,E)$. This is a fundamental graph problem with classical sequential algorithms that run in $\tilde{O}(n^3)$ and $\tilde{O}(mn)$ time where $n=|V|$ and $m=|E|$. In recent years this problem has received significant attention in the context of fine-grained sequential complexity as well as in the design of faster sequential approximation algorithms, though not much is known in the distributed CONGEST model. We present sublinear-round approximation algorithms for computing MWC in directed graphs, and weighted graphs. Our algorithms use a variety of techniques in non-trivial ways, such as in our approximate directed unweighted MWC algorithm that efficiently computes BFS from all vertices restricted to certain implicitly computed neighborhoods in sublinear rounds, and in our weighted approximation algorithms that use unweighted MWC algorithms on scaled graphs combined with a fast and streamlined method for computing multiple source approximate SSSP. We present $\tildeΩ(\sqrt{n})$ lower bounds for arbitrary constant factor approximation of MWC in directed graphs and undirected weighted graphs.

Improved Approximation Bounds for Minimum Weight Cycle in the CONGEST Model

TL;DR

This paper focuses on round complexity bounds for approximating MWC in the CONGEST model, and shows that for any constant , computing an -approximation of MWC requires rounds on weighted undirected graphs and on directed graphs, even if unweighted.

Abstract

Minimum Weight Cycle (MWC) is the problem of finding a simple cycle of minimum weight in a graph . This is a fundamental graph problem with classical sequential algorithms that run in and time where and . In recent years this problem has received significant attention in the context of fine-grained sequential complexity as well as in the design of faster sequential approximation algorithms, though not much is known in the distributed CONGEST model. We present sublinear-round approximation algorithms for computing MWC in directed graphs, and weighted graphs. Our algorithms use a variety of techniques in non-trivial ways, such as in our approximate directed unweighted MWC algorithm that efficiently computes BFS from all vertices restricted to certain implicitly computed neighborhoods in sublinear rounds, and in our weighted approximation algorithms that use unweighted MWC algorithms on scaled graphs combined with a fast and streamlined method for computing multiple source approximate SSSP. We present lower bounds for arbitrary constant factor approximation of MWC in directed graphs and undirected weighted graphs.
Paper Structure (31 sections, 16 theorems, 5 equations, 3 figures, 1 table, 6 algorithms)

This paper contains 31 sections, 16 theorems, 5 equations, 3 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our Results
  4. Directed Graphs.
  5. Undirected Unweighted Graphs.
  6. Undirected Weighted Graphs.
  7. Approximate k-source SSSP
  8. Significance of our Results
  9. Techniques
  10. Prior Work
  11. Sequential Minimum Weight Cycle
  12. Distrbuted Minimum Weight Cycle
  13. Distributed Girth
  14. Fixed-Length Cycle Detection
  15. CONGEST results for APSP and related problems
  16. ...and 16 more sections

Key Result

Theorem 1

Let $G=(V,E)$ be a directed graph. In the CONGEST model, for any constants $\epsilon > 0, \alpha\ge 2$:

Figures (3)

  • Figure 1: Directed MWC Lower Bound for $\alpha$-approximation
  • Figure 2: Undirected Weighted MWC Lower Bound for $\alpha$-approximation
  • Figure 3: Undirected Unweighted MWC Lower Bound for $2.5-\epsilon$-approximation

Theorems & Definitions (35)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2
  • Theorem 4
  • proof : Proof of Theorem \ref{['thm:dirlb']}.\ref{['thm:dirlb:alphalb']}
  • proof : Proof of Theorem \ref{['thm:undirwt']}.\ref{['thm:undirwt:alphalb']}
  • proof : Proof of Theorem \ref{['thm:undirunwt']}.\ref{['thm:undirunwt:alphalb']}
  • Lemma 5
  • ...and 25 more