On the Road to $\mathcal{PLS}$-Completeness: 8 Agents in a Singleton Congestion Game

Abstract

In this paper, we investigate the complexity of computing locally optimal solutions for Singleton Congestion Games (SCG) in the framework of $\mathcal{PLS}$, as defined in Johnson et al. [25]. Here, in an instance weighted agents choose links from a set of identical links. The cost of an agent is the load (the sum of the weights of the agents) on the link it chooses. The agents are selfish and try to minimize their individual cost. Agents may form arbitrary, non-fixed coalitions. The cost of a coalition is defined to be the maximum cost of its members. The potential function is defined as the lexicographical order of the agents’ cost. In each selfish step of a coalition, the potential function decreases. Thus, a local minimum is a Nash Equilibrium among coalitions of size at most k—an assignment where no coalition of size at most k has an incentive to unilaterally decrease its cost by switching to different links. The neighborhood of a feasible assignment (every agent chooses a link) are all assignments, where the cost of some arbitrary non-fixed coalition of at most k reallocating agents decreases. We call this problem SCG-(k) and show that SCG-(k) is $\mathcal{PLS}$-complete for k ≥ 8. On the other hand, for k = 1, it is well known that the solution computed by Graham’s LPT-algorithm [14,16,22] is locally optimal for SCG-(k).

We show our result by tight reduction from the MaxConstraintAssignment-problem (p,q,r)-MCA, which is an extension of Generalized Satisfiability to higher valued variables. Here, p is the maximum number of variables occurring in a constraint, q is the maximum number of appearances of a variable, and r is the valuedness of the variables.

To the best of our knowledge, SCG-(k) is the first problem, which is known to be solvable in polynomial time for a small neighborhood and $\mathcal{PLS}$-complete for a larger, but still constant neighborhood.

This work has been partially supported by the European Union within the Integrated Project IST-15964 “AEOLUS” and the German Science Foundation (DFG) Research Training Group GK-693 of the Paderborn Institute for Scientific Computation (PaSCo).