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

  • Authors:

  • Dominic Dumrauf;Burkhard Monien

  • Affiliations:

  • Paderborn Institute for Scientific Computation, University of Paderborn, Paderborn, Germany 33102 and Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, ...;Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, Paderborn, Germany 33102

  • Venue:
  • WINE '08 Proceedings of the 4th International Workshop on Internet and Network Economics
  • Year:
  • 2008

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Abstract

In this paper, we investigate the complexity of computinglocally optimal solutions for Singleton Congestion Games (SCG) inthe framework of $\mathcal{PLS}$, as defined in Johnson et al.[25]. Here, in an instance weighted agents choose links from a setof identical links. The cost of an agent is the load (the sum ofthe weights of the agents) on the link it chooses. The agents areselfish and try to minimize their individual cost. Agents may formarbitrary, non-fixed coalitions. The cost of a coalition is definedto be the maximum cost of its members. The potential function isdefined as the lexicographical order of the agents’ cost. Ineach selfish step of a coalition, the potential function decreases.Thus, a local minimum is a Nash Equilibrium among coalitions ofsize at most k—an assignment where no coalition of size atmost k has an incentive to unilaterally decrease its cost byswitching to different links. The neighborhood of a feasibleassignment (every agent chooses a link) are all assignments, wherethe cost of some arbitrary non-fixed coalition of at most kreallocating agents decreases. We call this problem SCG-(k) andshow that SCG-(k) is $\mathcal{PLS}$-complete for k ≥ 8.On the other hand, for k = 1, it is well known thatthe solution computed by Graham’s LPT-algorithm [14,16,22] islocally optimal for SCG-(k). We show our result by tight reductionfrom the MaxConstraintAssignment-problem (p,q,r)-MCA, which is anextension of Generalized Satisfiability to higher valued variables.Here, p is the maximum number of variables occurring in aconstraint, q is the maximum number of appearances of a variable,and r is the valuedness of the variables. To the best of ourknowledge, SCG-(k) is the first problem, which is known to besolvable in polynomial time for a small neighborhood and$\mathcal{PLS}$-complete for a larger, but still constantneighborhood.