Deterministic Even-Cycle Detection in Broadcast CONGEST

2024-12-15

arXiv.org

AI Summary

This paper presents the first deterministic optimal algorithm for $C_{2k}$-freeness—i.e., deciding whether an $n$-vertex graph contains an even cycle of length $2k$ ($k geq 2$)—in the Broadcast CONGEST model. To overcome long-standing reliance on randomization, it introduces a novel combinatorial framework combining parallel BFS exploration, deterministic path filtering, and a new notion of local density. Its core technical contribution is a structural density lemma for $C_{2k}$-free graphs, which yields an $O(n^{1-1/k})$-round complexity—tight up to a logarithmic factor for $k=2$, matching the state-of-the-art randomized algorithms, and uniformly applicable for all $k geq 2$. This is the first fully derandomized solution for even-cycle detection, resolving a fundamental bottleneck in the deterministic complexity of the problem.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We show that, for every $kgeq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for $k=2$ up to logarithmic factors thanks to the lower bound for $C_4$-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for $kin{3,4,5}$, and by Fraigniaud et al. [PODC 2024] for $kgeq 6$. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the"local density"of graphs without $2k$-cycles, which we believe is interesting on its own.

Problem

• Detect even-length cycles in Broadcast CONGEST model
• Develop deterministic algorithm for optimal round-complexity
• Bound local density in graphs without 2k-cycles

Innovation

• Deterministic algorithm for cycle detection
• Optimal round-complexity in CONGEST model
• Parallel BFS with deterministic path selection