# Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time

@article{Knemann2019ComputingTN,
title={Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time},
author={Jochen K{\"o}nemann and Kanstantsin Pashkovich and Justin Toth},
journal={ArXiv},
year={2019},
volume={abs/1803.03249}
}
• Published 2019
• Mathematics, Computer Science
• ArXiv
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 of [Kern and Paulusma, Mathematics of Operations Research, 2003].
11 Citations

#### Figures and Topics from this paper

Computing the Nucleolus of Weighted Voting Games in Pseudo-polynomial Time
We provide an algorithm for computing the nucleolus for an instance of a weighted voting game in pseudo-polynomial time. This resolves an open question posed by Elkind. et.al. 2007.
Arboricity Games: the Core and the Nucleolus
• Computer Science, Mathematics
• ArXiv
• 2020
The prime partition provides an analogous graph decomposition to [29, 30], which complements another line of research and introduces the arboricity game as a cooperative cost game defined on a graph. Expand
Simple Games Versus Weighted Voting Games: Bounding the Critical Threshold Value
• Computer Science
• ArXiv
• 2018
It is proved that for graphic simple games, that is, simple games in which every minimal winning coalition has size 2, computing α is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Expand
Approximate Core Allocations for Multiple Partners Matching Games
• Computer Science, Economics
• ArXiv
• 2021
This work studies approximate core allocations for the multiple partners matching game, and provides an LP-based mechanism guaranteeing that no coalition is paid less than 2 3 times the profit it makes on its own. Expand
G T ] 6 M ay 2 01 8 Simple Games versus Weighted Voting Games ⋆
• 2018
A simple game (N, v) is given by a set N of n players and a partition of 2 into a set L of losing coalitions L with value v(L) = 0 that is closed under taking subsets and a set W of winningExpand
Simple Games versus Weighted Voting Games
• Mathematics, Computer Science
• SAGT
• 2018
A simple game (N, v) is given by a set N of n players and a partition of $$2^N$$ into a set $$\mathcal {L}$$ of losing coalitions L with value $$v(L)=0$$ that is closed under taking subsets and a setExpand
Simple games versus weighted voting games: bounding the critical threshold value
• Computer Science, Mathematics
• Soc. Choice Welf.
• 2020
It is proved that for graphic simple games, that is, simple games in which every minimal winning coalition has size 2, the problem of computing α is NP -hard, but polynomial-time solvable if the underlying graph is bipartite. Expand
Computing Balanced Solutions for Large International Kidney Exchange Schemes
• Computer Science
• ArXiv
• 2021
The goal is to find a solution in the patient-donor compatibility graph that approaches the target allocation as closely as possible, to ensure long-term stability of the international pool of IKEPs. Expand
On the Complexity of Nucleolus Computation for Bipartite b-Matching Games
• Computer Science
• SAGT
• 2021
It is shown that computing the nucleolus of a simple b-matching game is NP-hard when b ≡ 3 even on bipartite graphs of maximum degree 7 and an efficient algorithm is described when a constant number of vertices satisfy bv = 2. Expand
Population Monotonicity in Matching Games
• Computer Science
• ArXiv
• 2021
This paper studies matching games and provides a necessary and sufficient characterization for the existence of population monotonic allocation schemes and implies that whether a matching game admits population Monotonic Allocation schemes can be determined efficiently. Expand

#### References

SHOWING 1-10 OF 60 REFERENCES
Note Computing the nucleolus of min-cost spanning tree games is NP-hard
• Mathematics, Computer Science
• Int. J. Game Theory
• 1998
It is proved that computing the nucleolus of minimum cost spanning tree games is in general NP-hard using a reduction from minimum cover problems. Expand
The nucleon of cooperative games and an algorithm for matching games
• Computer Science, Mathematics
• Math. Program.
• 1995
The nucleon may be viewed as the multiplicative analogue of Schmeidler's nucleolus and it is shown that the nucleon of (not necessarily bipartite) matching games can be computed in polynomial time. Expand
The nucleolus of balanced simple flow networks
• Mathematics, Computer Science
• Games Econ. Behav.
• 2006
This paper gives an algorithm for the nucleolus of simple flow games with directed and undirected, private as well as public arcs, under the condition that the flow game has a nonempty core.
Computing the Nucleolus of Matching, Cover and Clique Games
• Computer Science
• AAAI
• 2012
This work studies the computation of the nucleolus of a number of cooperative games, including fractional matching games and fractional edge cover games on general weighted graphs, as well as vertex cover games and clique games on weighted bipartite graphs. Expand
COMPUTATION OF THE KERNELS OF SIMPLE GAMES AND THE NUCLEOLUS OF N-PERSON GAMES.
Abstract : Methods for computing the kernel and nucleolus of simple games are described. Tables of results are given for weighted majority 7-person constant sum games, and 6-person superadditiveExpand
Matching Games: The Least Core and the Nucleolus
• Mathematics, Computer Science
• Math. Oper. Res.
• 2003
It is shown that the nucleolus of weighted matching games can be computed efficiently, based on an alternative characterization of the least core, which may be of independent interest. Expand
The kernel/nucleolus of a standard tree game
• Mathematics
• 1996
In this paper we characterize the nucleolus (which coincides with the kernel) of a tree enterprise. We also provide a new algorithm to compute it, which sheds light on its structure. We show that inExpand
Computing the nucleolus of some combinatorially-structured games
• Mathematics, Computer Science
• Math. Program.
• 2000
The ℬ-pren nucleolus is a straightforward generalization of the ordinary prenucleolus for a transferable utility game (N,v), whereℬ⊆2N means minimum cost spanning tree games. Expand
Note on the computational complexity of least core concepts for min-cost spanning tree games
• Mathematics, Computer Science
• Math. Methods Oper. Res.
• 2000
By a reduction from minimum cover problems, it is proved that computing an element in these least cores is in general NP-hard for minimum cost spanning tree games. Expand
On the computation of the nucleolus of a cooperative game
• Mathematics, Computer Science
• Int. J. Game Theory
• 2001
The algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel and computes the nucleolus of the game in the case where the pre Kernel contains exactly one core vector. Expand