Computational complexity of fixed points

Sikorski, K.

doi:10.1007/s11784-009-0128-3

Cited by 4 publications

(2 citation statements)

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“…For arbitrary maps with c > 1, exponential bounds on the query complexity of computing fixpoints are known [8,41]. In [6,44,72], algorithms for computing fixpoints for specialized maps such as weakly (c = 1) or strictly (c < 1) contracting maps are studied. For both cases, algorithms are known for the case of ℓ 2 and ℓ ∞ norms, both for absolute approximation (||x − x * || ≤ ǫ where x * is an exact fixpoint) and relative approximation (||x − f (x)|| ≤ ǫ).…”

Section: Related Workmentioning

confidence: 99%

Unique End of Potential Line

Fearnley¹,

Gordon²,

Mehta³

et al. 2018

Preprint

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This paper studies 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 UniqueEOPL to capture problems of this type. We first define a class that we call EOPL, which consists of all problems that can be reduced to EndOfPo-tentialLine. This problem merges the canonical PPAD-complete problem EndOfLine, with the canonical PLS-complete problem SinkOfDag, and so EndOfPotentialLine captures problems that can be solved by a line-following algorithm that also simultaneously decreases a potential function.PromiseUEOPL is a promise-subclass of EOPL in which the line in the EndOfPoten-tialLine instance is guaranteed to be unique via a promise. We turn this into a non-promise class UniqueEOPL, by adding an extra solution type to EndOfPotentialLine that captures any pair of points that are provably on two different lines.We show that UniqueEOPL ⊆ EOPL ⊆ CLS, and that all of our motivating problems are contained in UniqueEOPL: specifically USO, P-LCP, and finding a fixpoint of a Piecewise-Linear Contraction under an ℓ p -norm all lie in UniqueEOPL. Until now, USO was not even known to lie in PPAD or PLS. Our results also imply that parity games, mean-payoff games, discounted games, and simple-stochastic games lie in UniqueEOPL.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 UniqueEOPL, and we are also able to show that OPDC is UniqueEOPL-complete.Finally, using the insights from our reduction for Piecewise-Linear Contraction, we obtain the first polynomial-time 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 UniqueEOPL allows a technique of Aldous [2] to be applied, which in turn gives the fastest-known randomized algorithm for P-LCP.

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Section: Related Workmentioning

confidence: 99%

Unique End of Potential Line

Fearnley¹,

Gordon²,

Mehta³

et al. 2018

Preprint

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“…For arbitrary maps with c > 1, exponential bounds on the query complexity of computing fixpoints are known [7,33]. In [5,35,59], algorithms for computing fixpoints for specialized maps such as weakly (c = 1) or strictly (c < 1) contracting maps are studied. For both cases, algorithms are known for the case of ℓ 2 and ℓ ∞ norms, both for absolute approximation (||x − x * || ≤ ǫ where x * is an exact fixpoint) and relative approximation (||x − f (x)|| ≤ ǫ).…”

Section: Simple Stochastic Gamesmentioning

confidence: 99%

End of Potential Line

Fearnley,

Gordon,

Mehta

et al. 2018

Preprint

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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.

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