Simple 2-approximations for bad triangle transversals and some hardness results for related problems

TL;DR

This paper introduces simple, efficient 2-approximation algorithms for the bad triangle transversal problem in signed graphs, along with hardness results and relations to correlation clustering.

Contribution

It presents novel, faster 2-approximation algorithms for BTT, extends results to weighted cases, and establishes new hardness bounds and connections to correlation clustering.

Findings

Proposed simple 2-approximation algorithms for BTT.

Achieved a $(2+)$ approximation using LP and maximal edge-disjoint bad triangles.

Established NP-hardness of approximating BTT within factor 137/2136.

Abstract

Given a signed graph, the bad triangle transversal (BTT) problem asks to find the smallest number of edges that need to be removed such that the remaining graph does not have a triangle with exactly one negative edge (a bad triangle). We propose novel 2-approximations for this problem, which are much simpler and faster than a folklore adaptation of the 2-approximation by Krivelevich for finding a minimum triangle transversal in unsigned graphs. One of our algorithms also works for weighted BTT and for approximately optimal feasible solutions to the bad triangle cover LP. Using a recent result on approximating the bad triangle cover LP, we obtain a $(2+\epsilon)$ approximation in time almost equal to the time needed to find a maximal set of edge-disjoint bad triangles (which would give a standard 3-approximation). Additionally, several inapproximability results are provided. For complete signed graphs, we show that BTT is NP-hard to approximate with factor better than $\frac{2137}{2136}$. Our reduction also implies the same hardness result for related problems such as correlation clustering (cluster editing), cluster deletion and the min. strong triadic closure problem. On complete signed graphs, BTT is closely related to correlation clustering. We show that the correlation clustering optimum is at most $3/2$ times the BTT optimum, by describing a pivot procedure that transforms BTT solutions into clusters. This improves a result by Veldt, which states that their ratio is at most two.