Minimum-cost spanning tree games: Bird rule revisited

Abstract

Minimum-cost spanning tree (MCST) game is a classical model in the theory of cooperative games. The MCST game has been drawing continuous attention since it was raised by Claus and Kleitman in 1973. The MCST game is closely related to the MCST problem, a fundamental model in graph theory and combinatorial optimization, and has been widely applied in cost allocations for constructions of water networks, power networks, road networks and railway networks, etc. Due to its simplicity and intuitiveness, the Bird rule is one of the most famous solutions in the MCST game and has been extensively studied in the literature. Based on the linear programming formulation of the MCST problem by Edmonds, we provide a new equivalent closed-form formula for the Bird rule. We also study a generalization of the MCST game with weighted nodes. We show that the core is nonempty in the generalized model, and in particular, a modified Bird rule is proved to be still applicable.