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

2026-02-04

AI Summary

This study addresses the Bad Triangle Transversal (BTT) problem—deleting the minimum number of edges to eliminate all “bad triangles” (triangles with exactly one negative edge) in a signed graph. The problem is NP-hard, and this work proposes a simpler and more efficient 2-approximation algorithm, which is further extended to the weighted setting. By combining combinatorial optimization, linear programming rounding, and extremal graph theory, the authors obtain a (2+ε)-approximation algorithm whose runtime nearly matches that of finding a maximum edge-disjoint collection of bad triangles. Additionally, they establish a hardness of approximation threshold of 2137/2136 for BTT on complete signed graphs and improve the upper bound on the approximation ratio for a related clustering problem to 3/2.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

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.

Problem

• bad triangle transversal
• signed graph
• correlation clustering
• inapproximability
• triangle transversal

Innovation

• bad triangle transversal
• approximation algorithm
• correlation clustering
• inapproximability
• signed graph