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Theory of Computing Systems

, Volume 62, Issue 6, pp 1525–1552 | Cite as

Covering Triangles in Edge-Weighted Graphs

  • Xujin Chen
  • Zhuo Diao
  • Xiaodong Hu
  • Zhongzheng Tang
Article
  • 107 Downloads
Part of the following topical collections:
  1. Special Issue on Combinatorial Algorithms

Abstract

Let G = (V, E) be a simple graph and \(\mathbf {w}\in \mathbb {Z}^{E}_{>0}\) assign each edge eE a positive integer weight w(e). A subset of E that intersects every triangle of G is called a triangle cover of (G, w), and its weight is the total weight of its edges. A collection of triangles in G (repetition allowed) is called a triangle packing of (G, w) if each edge eE appears in at most w(e) members of the collection. Let τ t (G, w) and ν t (G, w) denote the minimum weight of a triangle cover and the maximum cardinality of a triangle packing of (G, w), respectively. Generalizing Tuza’s conjecture for unit weight, Chapuy et al. conjectured that τ t (G, w)/ν t (G, w) ≤ 2 holds for every simple graph G and every \(\mathbf {w}\in \mathbb {Z}^{E}_{>0}\). In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding triangle covers of small weights. These algorithms imply new sufficient conditions for the conjecture of Chapuy et al. More precisely, given (G, w), suppose that all edges of G are covered by the set \({\mathscr {T}}_{G}\) consisting of edge sets of triangles in G. Let \(|E|_{w}={\sum }_{e\in E}w(e)\) and \(|{\mathscr {T}}_{G}|_{w}={\sum }_{\{e,f,g\}\in {\mathscr {T}}_{G}}w(e)w(f)w(g)\) denote the weighted numbers of edges and triangles in (G, w), respectively. We show that a triangle cover of (G, w) of weight at most 2ν t (G, w) can be found in strongly polynomial time if one of the following conditions is satisfied: (i) \(\nu _{t}(G,\mathbf {w})/|{\mathscr {T}}_{G}|_{w}\ge \frac {1}{3}\), (ii) \(\nu _{t}(G,\mathbf {w})/|E|_{w}\ge \frac {1}{4}\), (iii) \(|E|_{w}/|{\mathscr {T}}_{G}|_{w}\ge 2\).

Keywords

Triangle cover Triangle packing Linear 3-uniform hypergraphs Combinatorial algorithms 

Notes

Acknowledgments

The authors are indebted to Dr. Gregory J. Puleo, Dr. Zbigniew Lonc, and two anonymous referees for their invaluable comments and suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Xujin Chen
    • 1
    • 2
  • Zhuo Diao
    • 3
  • Xiaodong Hu
    • 1
    • 2
  • Zhongzheng Tang
    • 1
    • 2
    • 4
  1. 1.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.School of Statistics and MathematicsCentral University of Finance and EconomicsBeijingChina
  4. 4.Department of Computer ScienceCity University of Hong Kong, HKSARKowlon TongChina

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