Optimization Method of Cross-monotonic Cost Sharing

In this paper, we studied upper bounds for the budget balance factor of cross-monotonic cost-sharing schemes for a variety of combinatorial optimization schemes and show stronger upper bounds for several combinatorial optimization problems using a novel technique based on the probabilistic method. Our techniques are quite general and may prove applicable to variety of other interesting games. First, we investigate the limitations imposed by the cross- monotonic property on cost-sharing schemes for several combinatorial optimization games including edge cover, vertex cover, set cover, metric facility location, maximum flow, arborescence packing, and maximum matching. Second, we develop a novel technique based on the probabilistic method for proving upper bounds on the budget-balance factor of cross-monotonic cost sharing schemes, deriving tight or nearly-tight bounds for the set cover and the vertex cover. We show that there is no cross-monotonic cost sharing scheme that recovers more than a third of the total cost. Finally, we present several trivial group-strategy proof mechanisms and study some of the axioms that can be added to the definition of group-strategy proof mechanisms to eliminate such trivial mechanisms .We show that there is no cross- monotonic cost sharing scheme that recovers more than a third of the total cost.