Optimal cost sharing for capacitated facility location games

We consider cost sharing for a class of facility location games, where the strategy spaceof each player consists of the bases of a player-specific matroid defined on the set of re-sources. We assume that resources have nondecreasing load-dependent costs and player-specific delays. Our model includes the important special case of capacitated facility loca-tion problems, where players have to jointly pay for opened facilities. The goal is to designcost sharing protocols so as to minimize the resulting price of anarchy and price of stabil-ity. We investigate two classes of protocols: basic protocols guarantee the existence of atleast one pure Nash equilibrium and separable protocols additionally require that the result-ing cost shares only depend on the set of players on a resource. We find optimal basic andseparable protocols that guarantee the price of stability/price of anarchy to grow logarithmi-cally/linearly in the number of players. These results extend our previous results (cf. [44]),where optimal basic and separable protocols were given for the case of symmetric matroidgames without delays.We finally study the complexity of computing optimal cost shares. We derive severalhardness results showing that optimal cost shares cannot be approximated in polynomialtime within a logarithmic factor in the number of players, unless P = NP. For a restrictedclass of problems that include the above hard instances, we devise an approximation algo-rithm matching the logarithmic bound.