Doubly-Efficient Pseudo-Deterministic Proofs

TL;DR

This paper introduces doubly-efficient pseudo-deterministic proofs for polynomial-time search problems, enabling quick verification of canonical solutions with polynomial prover runtime, especially for problems with known or conjectured lower bounds.

Contribution

It presents the first doubly-efficient pseudo-deterministic algorithms for several natural problems, including linear programming, 3-SUM, hitting set, Zero Weight Triangle, Orthogonal Vectors, and k-Clique.

Findings

Doubly-efficient pseudo-deterministic NP proof for linear programming, 3-SUM, and related problems.

Doubly-efficient pseudo-deterministic MA proof for Orthogonal Vectors and k-Clique.

Proves the existence of such proofs for problems with long-standing complexity conjectures.

Abstract

In [20] Goldwasser, Grossman and Holden introduced pseudo-deterministic interactive proofs for search problems where a powerful prover can convince a probabilistic polynomial time verifier that a solution to a search problem is canonical. They studied search problems for which polynomial time algorithms are not known and for which many solutions are possible. They showed that whereas there exists a constant round pseudo deterministic proof for graph isomorphism where the canonical solution is the lexicographically smallest isomorphism, the existence of pseudo-deterministic interactive proofs for NP-hard problems would imply the collapse of the polynomial time hierarchy. In this paper, we turn our attention to studying doubly-efficient pseudo-deterministic proofs for polynomial time search problems: pseudo-deterministic proofs with the extra requirement that the prover runtime is polynomial and the verifier runtime to verify that a solution is canonical is significantly lower than the complexity of finding any solution, canonical or otherwise. Naturally this question is particularly interesting for search problems for which a lower bound on its worst case complexity is known or has been widely conjectured. We show doubly-efficient pseudo-deterministic algorithms for a host of natural problems whose complexity has long been conjectured. In particular, we show a doubly efficient pseudo-deterministic NP proof for linear programming, 3-SUM and problems reducible to 3-SUM, the hitting set problem, and the Zero Weight Triangle problem and show a doubly-efficient pseudo-deterministic MA proof for the Orthogonal Vectors problem and the $k$-Clique problem.