Reduction of Ultrametric Minimum Cost Spanning Tree Games to Cost Allocation Games on Rooted Trees

Abstract

A minimum cQst spanning tree game is called ultrametric if the cost function on the edges of the underlying network is an ultrametric. We show that every ultrametric mlnimum cc/st, spanning tree game is reduced to a cost allocation game on a rooted tree. It follows that there exist･ O(n2) time algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametric minimum cost spanning tree games, where n is the number of players.

A review of cooperative rules and their associated algorithms for minimum-cost spanning tree problems

“…(Bergantiños and Vidal-Puga 2007a) A comparative of the Shapley value in the irreducible game (folk solution) and in the private game (Kar solution) can be found in Trudeau (2014b). Ando and Kato (2010) prove that the Shapley value in the irreducible game, as well as the egalitarian solution and the nucleolus, can be computed in time O(|N | 2 ). For irreducible cost matrices, Trudeau and Vidal-Puga (2020) show that the Shapley value, the nucleolus and the permutation-weighted average of extreme points of the core of v i coincide.…”

Computation of the Shapley value of minimum cost spanning tree games: #P-hardness and polynomial cases

“…Hence, it follows from Propositions 2.3, 4.3 and 2.5 that Φ(c) can be computed in O(n 3 ) time. However, it is possible to have an O(n 2 ) time algorithm for computing Φ(c) (see [5] and [2]). …”

“…We show that if the cost function of the given network is a subtree distance [16], which is a weaker notion of tree metric (see [21]), then the Shapley value of the associated game can be computed in O(n 4 ) time, where n is the number of players. This class of MCST games properly includes the formerly known subclass of MCST games for which there exists a polynomial time algorithm computing the Shapley value (see [5], [2]). …”