Effective Guessing Has Unlikely Consequences

TL;DR

This paper investigates the limitations of nondeterministic guessing strategies in speeding up deterministic computations, showing that certain hypothetical speedups would imply unlikely complexity class containments and providing improved bounds for SAT decision procedures.

Contribution

It establishes theoretical limitations on the effectiveness of nondeterministic guessing to accelerate deterministic algorithms, linking such speedups to unlikely complexity class inclusions.

Findings

Subpolynomial nondeterministic guessing cannot polynomially speed up deterministic computations.

Logarithmic speedup with nondeterminism implies SAT is in NTIME(n).

SAT can be decided in O(n log n) nondeterministic steps under certain encodings.

Abstract

A classic result of Paul, Pippenger, Szemer\'edi and Trotter states that DTIME(n) is strictly contained in NTIME(n). The natural question then arises: could DTIME(t(n)) be contained in NTIME(n) for some superlinear time-constructible function t(n)? If such a function t(n) does exist, then there also exist effective nondeterministic guessing strategies to speed up deterministic computations. In this work, we prove limitations on the effectiveness of nondeterministic guessing to speed up deterministic computations by showing that the existence of effective nondeterministic guessing strategies would have unlikely consequences. In particular, we show that if a subpolynomial amount of nondeterministic guessing could be used to speed up deterministic computation by a polynomial factor, then P is strictly contained in NTIME(n). Furthermore, even achieving a logarithmic speedup at the cost of making every step nondeterministic would show that SAT is in NTIME(n) under appropriate encodings. Of possibly independent interest, under such encodings we also show that SAT can be decided in O(n lg n) steps on a nondeterministic multitape Turing machine, improving on the well-known O(n(lg n)^c) bound for some constant but undetermined exponent c which is at least 1.