The complexity of Unique k-SAT: An Isolation Lemma for k-CNFs (original) (raw)

On the Complexity of Random Satisfiability Problems with Planted Solutions

Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15, 2015

Random satisfiability problems with planted solutions exhibit an intriguing complexity gap: for problems on n variables with k variables per constraint, after O(n log n) random clauses the planted assignment becomes the unique solution but the best-known algorithms need at least max{n r/2 , n log n} to efficiently identify it (or even one that is correlated with it), for clause distributions that are (r − 1)-wise independent (thus r can be as high as k). We show a nearly tight unconditional lower bound ofΩ(max{n r/2 , n log n}) clauses for any statistical algorithma restricted class of algorithms introduced in [41, 29] that covers most algorithmic approaches commonly used in theory and practice. We complement this with a nearly matching upper bound: a simple, iterative, statistical algorithm that usesÕ(n r/2 ) clauses and time linear in this to find the planted assignment with high probability. As known approaches for planted satisfiability problems (spectral, MCMC, gradient-based, etc.) all have statistical analogues, this provides a rigorous explanation of the large gap between the identifiability and algorithmic identifiability thresholds for random satisfiability problems with planted solutions.

On the complexity of SAT

… of Computer Science, 1999. 40th Annual …, 1999

We show that non-deterministic time N T IM E(n) is not contained in deterministic time n 2−ǫ and poly-logarithmic space, for any ǫ > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n 2−ǫ ) and polylogarithmic space. A similar result is presented for uniform circuits.

Randomized algorithms for 3-SAT

, Schöning proposed a simple yet efficient randomized algorithm for solving the k-SAT problem. In the case of 3-SAT, the algorithm has an expected running time of poly(n) · (4/3) n = O(1.334 n ). In this paper we present randomized algorithms and show that one of them has O(1.3302 n ) expected running time, improving Schöning's algorithm. (Note. At this point, the fastest randomized algorithm for 3-SAT is the one given by Iwama and Tamaki [IT03] that runs in O(1.324 n ).) *

On the satisfiability and maximum satisfiability of random 3-CNF formulas

1993

We analyze the pure literal rule heuristic for computing a satisfying assignment to a random 3-CNF formula with n variables. We show that the pure literal rule by itself finds satisfying assignments for almost all 3-CNF formulas with up to 1.63n clauses, but it fails for more than 1.7n clauses.

On the Complexity of SAT (Revised)

2008

We show 1 that non-deterministic time N T IM E(n) is not contained in deterministic time n √ 2− and polylogarithmic space, for any > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n √ 2− ) and poly-logarithmic space. A similar result is presented for uniform circuits.

Complexity and Algorithms for Well-Structured k-SAT Instances

Lecture Notes in Computer Science

This paper initiates the study of SAT instances of bounded diameter. The diameter of an ordered CNF formula is defined as the maximum difference between the index of the first and the last occurrence of a variable. We study the complexity of the satisfiability, the counting and the maximization problems for formulas of bounded diameter. We investigate the relation between the diameter of a formula, and the tree-width and the path-width of its corresponding incidence graph, and show that under highly parallel and efficient transformations, diameter and path-width are equal up to a constant factor. Our main technical contribution is that the computational complexity of SAT, Max-SAT, #SAT grows smoothly with the diameter (as a function of the number of variables). Our main focus is in providing space efficient and highly parallel algorithms, while the running time of our algorithms matches previously known results. Among others, we show NL-completeness of SAT and NC 2 algorithms for Max-SAT, #SAT when diameter is O(log n). Given the tree decomposition of a formula, we further improve on the space efficiency to decide SAT as asked by Alekhnovich and Razborov [1].

On the Random Satisfiable Process

Combinatorics, Probability and Computing, 2009

In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas. randomly permute all 2kbinomnk2^k\binom{n}{k}2kbinomnk possible clauses over the variables x1,. . .,xn, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if, after its addition, the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order).Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruciński and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erdős, Suen and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties have been studied, such as planarity and H-freeness. Thus our model is a natural extension of this approach t...

On the satisfiability of random regular signed SAT formulas

2011

Regular signed SAT is a variant of the well-known satisfiability problem in which the variables can take values in a fixed set V ⊂ [0, 1], and the literals have the form "x ≤ a" or "x ≥ a" instead of "x" or "x". We answer some open question regarding random regular signed k-SAT formulas: The probability that a random formula is satisfiable increases with |V |; there is a constant upper bound on the ratio m/n of clauses m over variables n, beyond which a random formula is asypmtotically almost never satisfied; for k = 2 and V = [0, 1], there is a phase transition at m/n = 2.