Fair cost-sharing methods for scheduling jobs on parallel machines

Abstract

a r t i c l e i n f o a b s t r a c tWe consider the problem of sharing the cost of scheduling n jobs on m parallel machines among a set of agents. In our setting, each agent owns exactly one job and the cost is given by the makespan of the computed assignment. We focus on α-budget-balanced crossmonotonic cost-sharing methods since they guarantee the two substantial mechanism properties α-budget-balance and group-strategyproofness and provide fair cost-shares. For identical jobs on related machines and for arbitrary jobs on identical machines, we give (m + 1)/(2m)-budget-balanced cross-monotonic cost-sharing methods and show that this is the best approximation possible. As our major result, we prove that the approximation factor for cross-monotonic cost-sharing methods is unbounded for arbitrary jobs and related machines. We therefore develop a cost-sharing method in the (m + 1)/(2m)-core, a weaker but also fair solution concept.

Efficient cost-sharing mechanisms for prize-collecting problems

“…Moulin's framework has been applied to several classical optimization problems such as submodular cost-sharing [31], fixed-tree multicast [3,14,15], minimum spanning tree [23,25], Steiner tree [23], Steiner forest [27], set cover [21], facility location [33], connected facility location [19,28,33], and machine scheduling [8,10]. Lower bounds on the budget balance factor that is achievable by a cross-monotonic cost-sharing mechanism were studied in [21,26,27].…”

“…We then use cross-monotonicity of ξ and the fact that β ≥ 1 to get (8). Inequality (9) uses the competitiveness of ξ, and the final inequality follows from the definition of Π 1 .…”

Quantifying inefficiency in cost-sharing mechanisms

“…Until very recently [33], the only general technique for designing mechanisms of this type was due to Moulin [35]. Researchers have developed numerous approximately budget-balanced Moulin mechanisms for cost-sharing problems arising from different combinatorial optimization problems, including fixed-tree multicast problems [2,15,16]; more general submodular problems [35,36]; scheduling problems [6,8]; network design problems [19,20,24,25,27,29,39]; facility location problems [30,39]; and various covering problems [12,23]. With one exception discussed below, none of these works provided any guarantees on the economic efficiency achieved by the proposed mechanisms.…”

“…However, it is not obvious how to use their efficiency loss measure to make comparisons between different cost-sharing problems. Additionally, the approach in [36] has not yet been extended beyond submodular cost-sharing problems, and most of the problems studied in the computer science literature fall outside of this class [6,8,19,20,23,24,25,27,29,30,39]. …”

“…(See Section 2 for formal definitions.) Until very recently, almost all approximately budget-balanced cost-sharing mechanisms were Moulin mechanisms [6,8,19,20,23,24,25,27,29,30,36,39], with the mechanisms of Devanur, Mihail, and Vazirani [12] forming a notable exception.…”