arXiv subjects

Daniel Ebert

Publications and source records attributed to Daniel Ebert.

2 recordsLinked to original sources

arXiv2026

Nucleolus Computation by Non-Zero-Constrained Optimization

We extend the list of games where the nucleolus is computable in polynomial time. Based on the classical MPS scheme, nucleolus computation can be reduced to the problem of finding a coalition with minimum excess that does not belong to a given linear subspace. We call this problem LSA-MinExcess, and show that it is equivalent to NZ-MinExcess: Given integral values per player, find a coalition with minimum excess whose player values do not sum up to $0$. Exploiting this representation, we prove that the nucleolus is computable in polynomial time for arboricity games, network strength games, and certain $b$-matching games. Along these lines, we show that for $b$-matching games with $b \leq 2$, LSA-MinExcess is polynomially equivalent to Shortest Non-Zero Cycle. Further, we prove that in general, linear subspace avoidance strictly increases the complexity of the minimum excess problem, even for monotone games. We still provide a reduction that trades linear subspace avoidance against an arbitrarily small approximation error. Finally, we show that the nucleolus is unstable in the following sense: A small change in the value function of the game can lead to a change in the nucleolus that is exponential in the number of players.

cs.GT↗

arXiv2025

Nucleolus, Happy Nucleolus, and Vehicle Routing

We study the recently introduced fair division concept of the happy nucleolus for cost allocation among players in a cooperative game, with special focus on its computation. The happy nucleolus applies the same fairness criterion as the well-established nucleolus but with reduced total value. Still, we show that the relation between the two concepts is quite involved, and intuitive properties do not hold - e.g., the entry of a player in the happy nucleolus can be larger than the entry of the same player in the nucleolus, even for monotone and subadditive games. This refutes conjectures of Meir, Rosenschein and Malizia (2011). Further, we study the separation problem of the linear programs appearing in the MPS scheme for computing the (happy) nucleolus. It includes linear subspace avoidance constraints, which can be handled efficiently for problems with a certain dynamic programming formulation due to K\"ohnemann and Toth (2020). We show how to get rid of these constraints for all monotone games if we allow for an arbitrarily small error of epsilon, thus conserving known approximation guarantees for the same problem without subspace avoidance. Finally, we focus on practical results at the example of vehicle routing games by designing an efficient heuristic based on our previous insights and past work, and demonstrate its power.

cs.GT↗