Beyond Distributed Subgraph Detection: Induced Subgraphs, Multicolored Problems and Graph Parameters

TL;DR

This paper explores the distributed complexity of induced subgraph detection in the CONGEST model, establishing lower bounds and efficient algorithms based on graph structural parameters like degeneracy and vertex cover.

Contribution

It provides new lower bounds and algorithms for induced subgraph detection in distributed settings, considering various graph classes and related problems.

Findings

Unconditional lower bounds for cycle and treewidth 2 pattern detection in general graphs.

Fast detection of induced paths in bounded degeneracy graphs.

Easy detection of induced subgraphs on graphs with bounded vertex cover.

Abstract

Subgraph detection has recently been one of the most studied problems in the CONGEST model of distributed computing. In this work, we study the distributed complexity of problems closely related to subgraph detection, mainly focusing on induced subgraph detection. The main line of this work presents lower bounds and parameterized algorithms w.r.t structural parameters of the input graph: -- On general graphs, we give unconditional lower bounds for induced detection of cycles and patterns of treewidth 2 in CONGEST. Moreover, by adapting reductions from centralized parameterized complexity, we prove lower bounds in CONGEST for detecting patterns with a 4-clique, and for induced path detection conditional on the hardness of triangle detection in the congested clique. -- On graphs of bounded degeneracy, we show that induced paths can be detected fast in CONGEST using techniques from parameterized algorithms, while detecting cycles and patterns of treewidth 2 is hard. -- On graphs of bounded vertex cover number, we show that induced subgraph detection is easy in CONGEST for any pattern graph. More specifically, we adapt a centralized parameterized algorithm for a more general maximum common induced subgraph detection problem to the distributed setting. In addition to these induced subgraph detection results, we study various related problems in the CONGEST and congested clique models, including for multicolored versions of subgraph-detection-like problems.