Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Complexity of Distributed Minimum Weight Cycle Approximation

Yi-Jun Chang, Yanyu Chen, Dipan Dey, Yonggang Jiang, Gopinath Mishra, Hung Thuan Nguyen, Mingyang Yang

Abstract

We investigate the \emph{minimum weight cycle (MWC)} problem in the $\mathsf{CONGEST}$ model of distributed computing. For undirected weighted graphs, we design a randomized algorithm that achieves a $(k+1)$-approximation, for any \emph{real} number $k \ge 1$. The round complexity of algorithm is \[ \tilde{O}\!\Big( n^{\frac{k+1}{2k+1}} + n^{\frac{1}{k}} + D\, n^{\frac{1}{2(2k+1)}} + D^{\frac{2}{5}} n^{\frac{2}{5}+\frac{1}{2(2k+1)}} \Big). \] where $n$ denotes the number of nodes and $D$ is the unweighted diameter of the graph. This result yields a smooth trade-off between approximation ratio and round complexity. In particular, when $k \geq 2$ and $D = \tilde{O}(n^{1/4})$, the bound simplifies to \[ \tilde{O}\!\left( n^{\frac{k+1}{2k+1}} \right) \] On the lower bound side, assuming the Erdős girth conjecture, we prove that for every \emph{integer} $k \ge 1$, any randomized $(k+1-ε)$-approximation algorithm for MWC requires \[ \tildeΩ\!\left( n^{\frac{k+1}{2k+1}} \right) \] rounds. This lower bound holds for both directed unweighted and undirected weighted graphs, and applies even to graphs with small diameter $D = Θ(\log n)$. Taken together, our upper and lower bounds \emph{match up to polylogarithmic factors} for graphs of sufficiently small diameter $D = \tilde{O}(n^{1/4})$ (when $k \geq 2$), yielding a nearly tight bound on the distributed complexity of the problem. Our results improve upon the previous state of the art: Manoharan and Ramachandran (PODC~2024) demonstrated a $(2+ε)$-approximation algorithm for undirected weighted graphs with round complexity $\tilde{O}(n^{2/3}+D)$, and proved that for any arbitrarily large number $α$, any $α$-approximation algorithm for directed unweighted or undirected weighted graphs requires $Ω(\sqrt{n}/\log n)$ rounds.

The Complexity of Distributed Minimum Weight Cycle Approximation

Abstract

We investigate the \emph{minimum weight cycle (MWC)} problem in the model of distributed computing. For undirected weighted graphs, we design a randomized algorithm that achieves a -approximation, for any \emph{real} number . The round complexity of algorithm is where denotes the number of nodes and is the unweighted diameter of the graph. This result yields a smooth trade-off between approximation ratio and round complexity. In particular, when and , the bound simplifies to On the lower bound side, assuming the Erdős girth conjecture, we prove that for every \emph{integer} , any randomized -approximation algorithm for MWC requires rounds. This lower bound holds for both directed unweighted and undirected weighted graphs, and applies even to graphs with small diameter . Taken together, our upper and lower bounds \emph{match up to polylogarithmic factors} for graphs of sufficiently small diameter (when ), yielding a nearly tight bound on the distributed complexity of the problem. Our results improve upon the previous state of the art: Manoharan and Ramachandran (PODC~2024) demonstrated a -approximation algorithm for undirected weighted graphs with round complexity , and proved that for any arbitrarily large number , any -approximation algorithm for directed unweighted or undirected weighted graphs requires rounds.

Paper Structure

This paper contains 76 sections, 37 theorems, 138 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prior Work.
  3. Our Contributions
  4. Technical Overview
  5. The Guiding Tradeoff.
  6. Lower Bound
  7. Comparison With Prior Work.
  8. Upper Bound
  9. Low-Diameter Decomposition.
  10. Short-Hop Cycles.
  11. Congestion--Dilation Tradeoff for SSSP.
  12. Long-Hop Cycles.
  13. Additional related work
  14. Connection to the Replacement Paths Problem
  15. Preliminaries
  16. ...and 61 more sections

Key Result

Theorem 1

Assuming the Erdős girth conjecture, we have the following lower bound. For any integer $p\geq 1$, $k\geq 1$ and $n\in \left\{d^{3p/2}\mid d\in \mathbb{Z}_{\geq 2} \right\}$, and for any $\varepsilon >0$, there exists a constant $\delta>0$ such that any $\delta$-error distributed algorithm for the $ rounds on some $\Theta(n)$-node directed (weighted or unweighted) or undirected weighted graph of d

Figures (3)

  • Figure 1: Tradeoff between approximation ratio and round complexity exponent (ignoring the polylogarithmic factor). Our results give a curve matches the lower bounds at discrete points when diameter is sufficiently small $D = \widetilde{O}(n^{1/4})$.
  • Figure 2: Lower Bound Graph $G(\gamma, k, d, p, H, x, y){}$ for MWC. The gray bipartite edges represent the base graph $H$ with high girth and the solid color edges represent the bipartite graph constructed by the algorithm depending on the inputs $x$ and $y$.
  • Figure 3: Squares nodes are sampled skeleton nodes

Theorems & Definitions (68)

  • Theorem 1: Lower bound
  • Theorem 2: Upper bound
  • proof
  • Corollary 1.1
  • Lemma 2.1: miller2013parallel
  • Theorem 3: ghaffari2015near
  • Definition 2.2: Representation of Skeleton Nodes
  • Lemma 2.4: Skeleton representation property
  • proof : Proof of \ref{['lem: skeleton nodes property']}
  • Conjecture 3.1: Erdős girth conjecture
  • ...and 58 more