# Unique End of Potential Line

@article{Fearnley2019UniqueEO,
title={Unique End of Potential Line},
author={John Fearnley and Spencer Gordon and Ruta Mehta and Rahul Savani},
journal={ArXiv},
year={2019},
volume={abs/1811.03841}
}
• Published 9 November 2018
• Mathematics
• ArXiv
This paper studies the complexity of problems in PPAD $\cap$ 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…
24 Citations

## Figures from this paper

The Complexity of Gradient Descent: CLS = PPAD $\cap$ PLS
• Mathematics
• 2020
It is shown that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain [0, 1] is PPAD∩PLS-complete, the first natural problem to be shown complete for this class.
Unique Sink Orientations of Grids is in Unique End of Potential Line
• Mathematics
• 2022
The complexity classes Unique End of Potential Line ( UEOPL ) and its promise version PromiseUEOPL were introduced in 2018 by Fearnly et al. [4]. PromiseUEOPL captures search problems where the
A New Combinatorial Property of Geometric Unique Sink Orientations
• Mathematics, Computer Science
ArXiv
• 2020
A new combinatorial property of the USOs that arise from symmetric P-LCP is established, which includes the US Os that arises from linear and simple convex quadratic programming.
Realizability Makes a Difference: A Complexity Gap for Sink-Finding in USOs
• Computer Science, Mathematics
ArXiv
• 2022
It is shown that in the realizable case O (log 2 n ) vertex evaluation queries suﬃce, while in general exactly n queries are needed, that the sink-ﬁnding problem might indeed be strictly easier on realizable USOs.
Computational Complexity of the α-Ham-Sandwich Problem
• Mathematics, Computer Science
ICALP
• 2020
It is shown that for the α-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL, the first non-trivial containment of the problem in a complexity class and places it in the company of several classic search problems.
Unique sink orientations for homogeneous linear inequalities and their alternative systems
It is not known, at the moment, whether there exists a strongly-polynomial algorithm for solving a system of linear inequalities. A noteworthy candidate is the simplex method, but despite a
Further Collapses in TFNP
• Mathematics
Electron. Colloquium Comput. Complex.
• 2022
We show EOPL = PLS ∩ PPAD . Here the class EOPL consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev ( SICOMP
Computational Complexity of the $\alpha$-Ham-Sandwich Problem.
• Computer Science, Mathematics
• 2020
This paper shows that for the $\alpha$-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL, the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the P-matrix linear complementarity problem.
CMSC 858 M : Fun with Hardness Spring
• Mathematics
• 2021
In this chapter, we define the class of counting problems #P, which contains all functions producing the number of solutions to some instance of a problem in NP. We analyze the hardness of this
The Complexity of Finding Fair Independent Sets in Cycles
The problem of finding a monochromatic edge in a Schrijver graph, given a succinct representation of a coloring that uses fewer colors than its chromatic number, is $\mathsf{PPA}$-complete as well.

## References

SHOWING 1-10 OF 98 REFERENCES
End of Potential Line
• Mathematics
ArXiv
• 2018
Using the insights from the reduction for PL-Contraction, the first polynomial-time algorithms for finding fixed points of contraction maps in fixed dimension for any $\ell_p$ norm are obtained, where previously such algorithms were only known for the $\ell-2$ and $\ll_\infty$ norms.
Continuous local search
• Mathematics
SODA '11
• 2011
We introduce CLS, for continuous local search, a class of polynomial-time checkable total functions that lies at the intersection of PPAD and PLS, and captures a particularly benign kind of local
The Linear Complementarity Problem , Lemke Algorithm , Perturbation , and the Complexity Class PPAD
• Mathematics
• 2011
We present a single sufficient condition for the processability of the Lemke algorithm for semimonotone Linear Complementarity problems (LCP) which unifies several sufficient conditions for a number
CLS: New Problems and Completeness
• Computer Science
ArXiv
• 2017
EndOfPotentialLine is introduced, which captures aspects of PPAD and PLS directly via a monotonic directed path, and is a likely candidate to capture the exact complexity of PLCP; the structure of Lemke-Howson 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.
Simple Local Search Problems That are Hard to Solve
• Computer Science
SIAM J. Comput.
• 1991
It is shown here that several natural, simple local search problems are PLS-complete, and thus just as hard.
An interior point potential reduction algorithm for the linear complementarity problem
• Mathematics
Math. Program.
• 1992
It is shown that whenM is positive semi-definite, the choice ofρ = 2n+ $$sqrt {2n}$$ yields a polynomial-time algorithm that covers the convex quadratic minimization problem.
Digraph Models of Bard-Type Algorithms for the Linear Complementarity Problem
• Mathematics, Computer Science
Math. Oper. Res.
• 1978
These digraphs show that such algorithms based on complementary pivoting for solving the linear complementarity problem can cycle even for symmetric, positive deFinite M, and provide some insight into the algorithms' behavior.
The complexity of interior point methods for solving discounted turn-based stochastic games
• Computer Science
CiE
• 2013
It is shown that for 2TBSGs with n states and discount factor γ, the lower bounds for κ, − δ, and 1/θ are all obtained using the same family of deterministic games.
How easy is local search?
• Computer Science
26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
• 1985
On the Complexity of Nash Equilibria and Other Fixed Points
• Mathematics
SIAM J. Comput.
• 2010
It is shown that the (exact or approximate) computation of Nash equilibria for 3 or more players is complete for FIXP, which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits (straight line programs) over basis with rational constants.