Generating Paths and Cuts in Multi-pole (Di)graphs

Boros, Endre; Elbassioni, Khaled; Gurvich, Vladimir; Khachiyan, Leonid; Makino, Kazuhisa

doi:10.1007/978-3-540-28629-5_21

Cited by 6 publications

(3 citation statements)

References 13 publications

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“…As shown in [1] path conjunctions can be generated in incremental polynomial time. Here we show that the X − e + Y method provides a simple alternative algorithm.…”

Section: Proposition 8 Assume That There Is a Procedures That Outputs mentioning

confidence: 99%

Generating Cut Conjunctions in Graphs and Related Problems

et al. 2007

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Let G = (V , E) be an undirected graph, and let B ⊆ V × V be a collection of vertex pairs. We give an incremental polynomial time algorithm to generate all minimal edge sets X ⊆ E such that every pair (s, t) ∈ B of vertices is disconnected in (V , E X), generalizing well-known efficient algorithms for generating all minimal s-t cuts, for a given pair s, t of vertices. We also present an incremental polynomial time algorithm for generating all minimal subsets X ⊆ E such that no (s, t) ∈ B is This research was partially supported by the National Science Foundation (Grant IIS-0118635), by DIMACS, the National Science Foundation's Center for Discrete Mathematics and Theoretical Computer Science, and by the Scientific Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan. Our friend and colleague, Leonid Khachiyan passed away with tragic suddenness while we were preparing this manuscript. (V , X ∪ B). Both above problems are special cases of a more general problem that we call generating cut conjunctions for matroids: given a matroid M on ground set S = E ∪ B, generate all minimal subsets X ⊆ E such that no element b ∈ B is spanned by E X. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V , E ∪ B), the more general problem of generating cut conjunctions for vectorial matroids turns out to be NP-hard.

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“…As shown in [1] path conjunctions can be generated in incremental polynomial time. Here we show that the X − e + Y method provides a simple alternative algorithm.…”

Section: Proposition 8 Assume That There Is a Procedures That Outputs mentioning

confidence: 99%

Generating Cut Conjunctions in Graphs and Related Problems

et al. 2007

Self Cite

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show abstract

“…As an illustration for the X−e+Y method, consider the path conjunction problem [3]: given an undirected graph G = (V, E) and a collection of k vertex pairs s i , t i ∈ V , enumerate all minimal edge sets X ⊆ E such that for all i = 1, . .…”

Section: Proposition 3 For Any Monotone Boolean Function π : 2 E → {mentioning

confidence: 99%

“…Hence the X − e + Y method reduces the path conjunction enumeration problem to the enumeration of all u-v paths in G , which can be done via backtracking [8] incrementally efficiently. Thus by Proposition 4 enumerating all minimal path conjunctions can be done in incremental polynomial time [3].…”

Section: Proposition 3 For Any Monotone Boolean Function π : 2 E → {mentioning

confidence: 99%

Generating Cut Conjunctions and Bridge Avoiding Extensions in Graphs

Khachiyan

Boros

Borys

et al. 2005

Algorithms and Computation

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Abstract. Let G = (V, E) be an undirected graph, and let B ⊆ V × V be a collection of vertex pairs. We give an incremental polynomial time algorithm to enumerate all minimal edge sets X ⊆ E such that every vertex pair (s, t) ∈ B is disconnected in (V, E X), generalizing wellknown efficient algorithms for enumerating all minimal s-t cuts, for a given pair s, t ∈ V of vertices. We also present an incremental polynomial time algorithm for enumerating all minimal subsets X ⊆ E such that no (s, t) ∈ B is a bridge in (V, X ∪ B). These two enumeration problems are special cases of the more general cut conjunction problem in matroids: given a matroid M on ground set S = E ∪ B, enumerate all minimal subsets X ⊆ E such that no element b ∈ B is spanned by E X. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V, E ∪B), the enumeration of cut conjunctions for vectorial matroids turns out to be NP-hard.

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Scientific contributions of Leo Khachiyan (a short overview)

Boros

Gurvich

2008

Discrete Applied Mathematics

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Generating Paths and Cuts in Multi-pole (Di)graphs

Cited by 6 publications

References 13 publications

Generating Cut Conjunctions in Graphs and Related Problems

Generating Cut Conjunctions in Graphs and Related Problems

Generating Cut Conjunctions and Bridge Avoiding Extensions in Graphs

Scientific contributions of Leo Khachiyan (a short overview)

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