Abstract

The complexity class CLS was proposed by Daskalakis and Papadimitriou in 2011 to understand the complexity of important NP search problems that admit both path following and potential optimizing algorithms. Here we identify a subclass of CLS – called UniqueEOPL – that applies a more specific combinatorial principle that guarantees unique solutions. We show that UniqueEOPL contains several important problems such as the P-matrix Linear Complementarity Problem, finding fixed points of Contraction Maps, and solving Unique Sink Orientations (USOs). We identify a problem – closely related to solving contraction maps and USOs – that is complete for UniqueEOPL.

Highlights

  • The complexity class TFNP contains search problems that are guaranteed to have a solution, and whose solutions can be verified in polynomial time [44]

  • It has attracted intense attention in the past decade, culminating in a series of papers showing that the problem of computing a Nash-equilibrium in two-player games is PPAD-complete [10, 13], and more recently a conditional lower bound that rules out a PTAS for the problem [52]

  • We provide a complete problem for UniqueEOPL, called One-Permutation Discrete Contraction (OPDC)

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Summary

Introduction

The complexity class TFNP contains search problems that are guaranteed to have a solution, and whose solutions can be verified in polynomial time [44]. Daskalakis and Papadimitriou [14] observed that several prominent total function problems for which no polynomial-time algorithms are known lie in PPAD ∩ PLS. We show that all of our motivating problems – USO, P-LCP, and finding a fixpoint of a Piecewise-Linear Contraction under an p-norm – are contained in UniqueEOPL (PromiseUEOPL for the promise versions) via promise-preserving reductions. A polynomial time algorithm for finding an approximate fixpoint of a contraction map in constant dimension can be obtained through a reduction to the Tarski fixpoint problem [51]. Our reduction inserts new vertices into the instance to satisfy this property

Promise problems with unique solutions
One-Permutation Discrete Contraction is UniqueEOPL-complete
UniqueEOPL containment results
New algorithms
Conjectures and conclusions
Full Text
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