
Revised: July 14, 2013
Published: October 8, 2014
Abstract: [Plain Text Version]
Alekhnovich and Razborov (2002) presented an algorithm that solves SAT on instances \phi of size n and tree-width \tw(\phi), using time and space bounded by 2^{O(\tw(\phi))}n^{O(1)}. Although several follow-up works appeared over the last decade, the first open question of Alekhnovich and Razborov remained essentially unresolved: Can one check satisfiability of formulas with small tree-width in polynomial space and time as above? We essentially resolve this question, by (1) giving a polynomial space algorithm with a slightly worse run-time, (2) providing a complexity-theoretic characterization of bounded tree-width SAT, which strongly suggests that no polynomial-space algorithm can run significantly faster, and (3) presenting a spectrum of algorithms trading off time for space, between our PSPACE algorithm and the fastest known algorithm.
First, we give a simple algorithm that runs in polynomial space and achieves run-time 3^{\tw(\phi)\log n}n^{O(1)}, which approaches the run-time of Alekhnovich and Razborov (2002), but has an additional \log n factor in the exponent. Then, we conjecture that this annoying \log n factor is in general unavoidable.
Our negative results show our conjecture true if one believes a well-known complexity assumption, which is the SC \neq NC conjecture and its scaled variants. Technically, we base our result on the following lemma. For arbitrary k, SAT of tree-width \log^k n is complete for the class of problems computed by circuits of logarithmic depth, semi-unbounded fan-in and size 2^{O(\log^k n)} (SAC^1 when k=1). Problems in this class can be solved simultaneously in time-space (2^{O(\log^{k+1}n)}, O(\log^{k+1}n)), and also in (2^{O(\log^k n)}, 2^{O(\log^k n)}). Then, we show that our conjecture (for SAT instances with poly-log tree-width) is equivalent to the question of whether the small-space simulation of semi-unbounded circuit classes can be sped up without incurring a large space penalty. This is a recasting of the conjecture that SAC^1 (and even its subclass NL) is not contained in SC.
Although we cannot hope for an improvement asymptotically in the exponent of time and space, we introduce a new algorithmic technique which trades constants in the exponents: for each \varepsilon with 0 < \varepsilon <1, we give an algorithm in time-space \big( 3^{1.441(1-\varepsilon)\tw(\phi)\log{|\phi|}}|\phi|^{O(1)},\; 2^{2\varepsilon\tw(\phi)}|\phi|^{O(1)} \big)\,. We systematically study the limitations of our technique for trading off time and space, and we show that our bounds are the best achievable using this technique.