The nucleolus of balanced simple flow networks

Potters, J.A.M.; Reijnierse, Hans; Biswas, Amit K.

doi:10.1016/j.geb.2004.08.008

Cited by 33 publications

(22 citation statements)

References 8 publications

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“…This enables one to disregard all coalitions not in D ∪ R ∪ 0 ∪ 1 . Because the proof for the nucleon in §2.3 goes along the same lines and the result for the nucleon is new, we leave out the proof details of Theorem 2.3, which has been shown in Deng et al [4] and Potters et al [25] as well. INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).…”

Section: The Nucleolus Of Simple Flow Games Given An Allocation X ∈mentioning

confidence: 99%

“…Fourteen years later, both Deng et al [4] and Potters et al [25] independently show that the nucleolus of a simple flow game can be computed in polynomial time. Both papers use the ellipsoid method, as opposed to the approach in Granot and Granot [15] that is based on removing redundant s-t path constraints in the sequence of linear programs that determine the nucleolus (we will explain this sequence of linear programs later).…”

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

“…Both papers use the ellipsoid method, as opposed to the approach in Granot and Granot [15] that is based on removing redundant s-t path constraints in the sequence of linear programs that determine the nucleolus (we will explain this sequence of linear programs later). Potters et al [25] also make use of a polynomial description of the core in terms of so-called potential functions defined on a (modified) network.…”

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

“…We would like to emphasize that our method does not rely on the ellipsoid method, as opposed to the "dual" approach in Deng et al [4], Potters et al [25] for computing the nucleolus in polynomial time, and (hopefully) provides some additional insight into the structure of the problem.…”

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

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On the Core and f-Nucleolus of Flow Games

Kern

Paulusma

2009

Mathematics of OR

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at https://www.informs.org/. show that the nucleolus of simple flow games (where all edge capacities are equal to one) can be computed in polynomial time. Our main result is a combinatorial method based on eliminating redundant s-t path constraints such that only a polynomial number of constraints remains. This leads to efficient algorithms for computing the core, nucleolus, and nucleon of simple flow games. Deng et al. also prove that computing the nucleolus for (general) flow games is NP-hard. We generalize this by proving that computing the f -nucleolus of flow games is NP-hard for all priority functions f that satisfy f A > 0 for all coalitions A with worth v A > 0 (so, including the priority functions corresponding to the nucleolus, nucleon, and per-capita nucleolus). MATHEMATICS OF OPERATIONS RESEARCHVol. 34, No. 4, November 2009, pp. 981-991 issn 0364-765X eissn 1526-5471 09 3404 0981 informs ® doi 10.1287

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Section: The Nucleolus Of Simple Flow Games Given An Allocation X ∈mentioning

confidence: 99%

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

See 2 more Smart Citations

On the Core and f-Nucleolus of Flow Games

Kern

Paulusma

2009

Mathematics of OR

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“…The first polynomial-time algorithm computing the nucleolus was proposed by Megiddo [18] for tree enterprises. Later on, several polynomial-time algorithms were proposed [23,14,12,5,21,9,1,3,2,4,19,11,16]. On the other hand, NP-hardness results for computing the nucleolus were proved in the papers [5,10,8].…”

Section: Introductionmentioning

confidence: 99%

The nucleolus of arborescence games in directed acyclic graphs

Kamiyama

2015

Operations Research Letters

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Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time

Könemann

Pashkovich

Toth

2019

Integer Programming and Combinatorial Optimization

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We provide an efficient algorithm for computing the nucleolus for an instance of a weighted cooperative matching game. This resolves a long-standing open question posed in [Faigle, Kern, Fekete, Hochstättler, Mathematical Programming, 1998].Let ε * be the optimum value of (P ), and define P (ε * ) to be the set of allocations x such that (x, ε * ) is feasible for (P ). The set P (ε * ) is known as the leastcore [39] of the given cooperative game, and the special case when ε * = 0, P (0) is the well-known core [25] of (V, ν). Intuitively, allocations in the core describe payoffs in which no coalition of players could profitably deviate from the grand coalition V . Why stop at maximizing the bottleneck excess? Consider an allocation which, subject to maximizing the smallest excess, maximizes the second smallest excess, and subject to that maximizes the third smallest excess, and so on. This process of successively optimizing the excess of the worst-off coalitions yields our primary object of interest, the nucleolus. For an allocation x ∈ R V , let θ(x) ∈ R 2 V −2 be the vector obtained by sorting the list of excess values x(S) − ν(S) for any ∅ = S ⊂ V in non-decreasing order 3 . The nucleolus, denoted η(V, ν) and defined by Schmeidler [45], is the unique allocation that lexicographically maximizes θ(x): η(V, ν) := arg lex max{θ(x) : x ∈ P (ε * )}.We refer the reader to Appendix B for an example instance of the weighted matching game with its nucleolus. We now have sufficient terminology to state our main result:Theorem 1. Given a graph G = (V, E) and weights w : E → R, the nucleolus η(V, ν) of the corresponding weighted matching game can be computed in polynomial time.

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The nucleolus of balanced simple flow networks

Cited by 33 publications

References 8 publications

On the Core and f-Nucleolus of Flow Games

On the Core and f-Nucleolus of Flow Games

The nucleolus of arborescence games in directed acyclic graphs

Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time

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