CLS: New Problems and Completeness

The complexity class CLS was introduced by Daskalakis and Papadimitriou in [9] with the goal of capturing the complexity of some well-known problems in PPAD ∩ PLS that have resisted, in some cases for decades, attempts to put them in polynomial time. No complete problem was known for CLS, and in [9], the problems Contraction, i.e., the problem of finding an approximate fixpoint of a contraction map, and P-LCP, i.e., the problem of solving a P-matrix Linear Complementarity Problem, were identified as prime candidates. First, we present a new CLS-complete problem MetametricContraction, which is closely related to the Contraction. Second, we introduce EndOfPotentialLine, which captures aspects of PPAD and PLS directly via a monotonic directed path, and show that EndOfPotentialLine is in CLS via a two-way reduction to EndOfMeteredLine. The latter was defined in [18] to keep track of how far a vertex is on the PPAD path via a restricted potential function. Third, we reduce P-LCP to EndOfPotentialLine, thus making EndOfPotentialLine and EndOfMeteredLine at least as likely to be hard for CLS as P-LCP. This last result leverages the monotonic structure of Lemke paths for P-LCP problems, making EndOfPotentialLine a likely candidate to capture the exact complexity of P-LCP; we note that the structure of LemkeHowson paths for finding a Nash equilibrium in a two-player game very directly motivated the definition of the complexity class PPAD, which eventually ended up capturing this problem’s complexity exactly. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes