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Covering Triangles in Edge-Weighted Graphs
Chen, Xujin1,2; Diao, Zhuo3; Hu, Xiaodong1,2; Tang, Zhongzheng1,2,4
AbstractLet G = (V, E) be a simple graph and assign each edge e a E 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 e a E appears in at most w(e) members of the collection. Let tau (t) (G, w) and nu (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 tau (t) (G, w)/nu (t) (G, w) ae 2 holds for every simple graph G and every . 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 consisting of edge sets of triangles in G. Let and 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 2 nu (t) (G, w) can be found in strongly polynomial time if one of the following conditions is satisfied: (i).t(G, w)/vertical bar TG vertical bar w >= 1/3, (ii).t(G, w)/vertical bar E vertical bar w >= 1/4, (iii) vertical bar E vertical bar(w)/vertical bar TG vertical bar(w) >= 2.
KeywordTriangle cover Triangle packing Linear 3-uniform hypergraphs Combinatorial algorithms
WOS Research AreaComputer Science ; Mathematics
WOS SubjectComputer Science, Theory & Methods ; Mathematics
WOS IDWOS:000434050300010
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Document Type期刊论文
Corresponding AuthorDiao, Zhuo
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Beijing, Peoples R China
2.Univ Chinese Acad Sci, Sch Math Sci, Beijing, Peoples R China
3.Cent Univ Finance & Econ, Sch Stat & Math, Beijing, Peoples R China
4.City Univ Hong Kong, Dept Comp Sci, Hksar, Kowlon Tong, Peoples R China
Recommended Citation
GB/T 7714
Chen, Xujin,Diao, Zhuo,Hu, Xiaodong,et al. Covering Triangles in Edge-Weighted Graphs[J]. THEORY OF COMPUTING SYSTEMS,2018,62(6):1525-1552.
APA Chen, Xujin,Diao, Zhuo,Hu, Xiaodong,&Tang, Zhongzheng.(2018).Covering Triangles in Edge-Weighted Graphs.THEORY OF COMPUTING SYSTEMS,62(6),1525-1552.
MLA Chen, Xujin,et al."Covering Triangles in Edge-Weighted Graphs".THEORY OF COMPUTING SYSTEMS 62.6(2018):1525-1552.
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