End of Potential Line

Abstract

We introduce the problem EndOfPotentialLine and the corresponding complexity class EOPL of all problems that can be reduced to it in polynomial time. This class captures problems that admit a single combinatorial proof of their joint membership in the complexity classes PPAD of fixpoint problems and PLS of local search problems. EOPL is a combinatorially-defined alternative to the class CLS (for Continuous Local Search), which was introduced in [16] 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. Two of these are Contraction, the problem of finding a fixpoint of a contraction map, and P-LCP, the problem of solving a P-matrix Linear Complementarity Problem.We show that EndOfPotentialLine is in CLS via a two-way reduction to EndOfMe-teredLine. The latter was defined in [36] to show query and cryptographic lower bounds for CLS. Our two main results are to show that both PL-Contraction (Piecewise-Linear Contraction) and P-LCP are in EOPL. Our reductions imply that the promise versions of PL-Contraction and P-LCP are in the promise class UniqueEOPL, which corresponds to the case of a single potential line. This also shows that simple-stochastic, discounted, mean-payoff, and parity games are in EOPL.Using the insights from our reduction for PL-Contraction, we obtain the first polynomialtime algorithms for finding fixed points of contraction maps in fixed dimension for any ℓ p norm, where previously such algorithms were only known for the ℓ 2 and ℓ ∞ norms. Our reduction from P-LCP to EndOfPotentialLine allows a technique of Aldous to be applied, which in turn gives the fastest-known randomized algorithm for P-LCP.

The complexity of splitting necklaces and bisecting ham sandwiches

“…In a recent paper, Sotiraki et al [67] identified the first natural problem for the class PPP, the class of problems whose totality is established by an argument based on the pigeonhole principle. For the class CLS, both Daskalakis et al [21] and Fearnley et al [27] identified complete problems (two versions of the Contraction Map problem, where a metric or a meta-metric are given as part of the input). In the latter paper, the authors define a new class, namely EOPL (for "End of Potential Line"), and show that it is a subclass of CLS.…”

“…Furthermore, they show that two well-known problems in CLS, the P-Matrix Linear Complementarity Problem (P-LCP), and finding a fixpoint of a piecewise-linear contraction map (PL-Contraction) belong to the class. The End of Potential Line problem of [27] is closely related to the End of Metered Line of [39].…”

The Hairy Ball Problem is PPAD-Complete

“…Its corresponding computational problem Contraction, is to find a fixed point of a given contraction map. Some versions of Contraction have been shown complete for CLS, a subclass of PPAD [11,12,16]. The search for Brouwer fixpoints (including discretised versions of Brouwer functions) is PPAD-complete for most variants of the problem [29,7], which is why we say the HBT is "Brouwer-like".…”