Finding Triangles or Independent Sets; and Other Dual Pair Approximations

TL;DR

This paper introduces new algorithms for triangle detection and related problems, demonstrating faster methods for approximating independent sets and graph coloring, and explores the dual pair approximation concept to unify these approaches.

Contribution

It presents novel randomized algorithms for triangle detection and approximation problems, and introduces the dual pair approximation framework for related graph problems.

Findings

A simple randomized algorithm for triangle detection.

Faster algorithms for approximating independent sets and graph coloring.

Introduction of the dual pair approximation concept.

Abstract

We revisit the algorithmic problem of finding a triangle in a graph (\textsc{Triangle Detection}), and examine its relation to other problems such as \textsc{3Sum}, \textsc{Independent Set}, and \textsc{Graph Coloring}. We obtain several new algorithms: \smallskip (I) A simple randomized algorithm for finding a triangle in a graph. As an application, we study the range of a conjecture of P\v{a}tra\c{s}cu (2010) regarding the triangle detection problem. \smallskip (II) An algorithm which given a graph $G=(V,E)$ performs one of the following tasks in $O(m+n)$ (ie, linear) time: (i)~compute a $\Omega(1/\sqrt{n})$-approximation of a maximum independent set in $G$ or (ii)~find a triangle in $G$. The run-time is faster than that for any previous method for each of these tasks. \smallskip (III) An algorithm which given a graph $G=(V,E)$ performs one of the following tasks in $O(m+n^{3/2})$ time: (i)~compute an $\sqrt{n}$-approximation for \textsc{Graph Coloring} of $G$ or (ii)~find a triangle in $G$. The run-time is faster than that for any previous method for each of these tasks on dense graphs, with $m =\omega(n^{9/8})$. \smallskip (IV) The second and third results suggest the following broader research direction: if it is difficult to find (A) or (B) separately, can one find one of the two efficiently? This motivates the \emph{dual pair} concept we introduce. We discuss and provide several instances of dual-pair approximation.