Unique End of Potential Line

We study the complexity of problems in PPAD ∩ PLS that have unique solutions. Three well-known examples of such problems are the problem of finding a fixpoint of a contraction map, finding the unique sink of a Unique Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem (P-LCP). Each of these are promise-problems, and when the promise holds, they always possess unique solutions.

We define the complexity class UEOPL to capture problems of this type. We show that UEOPL ⊆ CLS, and that all of our motivating problems are contained in UEOPL: specifically USO, P-LCP, and finding a fixpoint of a Piecewise-Linear Contraction under an ℓp-norm all lie in UEOPL. Our results also imply that parity games, mean-payoff games, discounted games, and simple-stochastic games lie in UEOPL.

All of our containment results are proved via a reduction to a problem that we call One-Permutation Discrete Contraction (OPDC). This problem is motivated by a discretized version of contraction, but it is also closely related to the USO problem. We show that OPDC lies in UEOPL, and we are also able to show that OPDC is UEOPL-complete.

Based on joint work with Spencer Gordon, Ruta Mehta, and Rahul Savani. https://arxiv.org/abs/1811.03841