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

Florian Adriaens, Nikolaj tatti

TL;DR

This work proposes novel 2-approximations for the bad triangle transversal 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.

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 $(2+ε)$ 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.

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

TL;DR

This work proposes novel 2-approximations for the bad triangle transversal 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.

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 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 . 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 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.
Paper Structure (14 sections, 16 theorems, 21 equations, 3 figures, 3 tables, 4 algorithms)

This paper contains 14 sections, 16 theorems, 21 equations, 3 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Results for general graphs
  3. Results for complete graphs
  4. Connections to STC+
  5. Connections to CC
  6. Recent CC algorithms based on BTT
  7. Other related work
  8. Approximation Algorithms
  9. Preliminaries
  10. BTT algorithms
  11. Hardness results
  12. Complete signed graphs
  13. From Covers to Clusters
  14. Conclusions

Key Result

Theorem 1

For complete signed graphs, there exists a randomized $(2+\epsilon)$-approximation algorithm (in expectation) for BTT in $\widetilde{\mathcal{O}}\xspace(\epsilon^{-7}m^{3/2})$ time, where $m$ is the number of positive edges. The algorithm can be derandomized in an additional $\widetilde{\mathcal{O}}

Figures (3)

  • Figure 1: Removing edges from an optimal BTT solution can lead to a bad cycle. Dashed edges are negative edges, all other pairs are connected by a positive edge (not drawn).
  • Figure 2: Part of the graph $G$ constructed from an instance of Theorem \ref{['thm:chlebik']}, if it contains a clause $C_\ell = \overline{x} \lor y$. The left (resp. right) hexagram is the node gadget corresponding to variable $x$ (resp. $y$). The $x_i$ nodes are the crown nodes of the hexagram of $x$, and similarly for $y_i$. The clause edges corresponding to clause $C_\ell$ are $(x_1,c_\ell)$ and $(c_\ell,y_6)$.
  • Figure 3: Example to illustrate case (b) in the proof of Lemma \ref{['lem:biggeropt']}.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 19 more