# Satisfiability Coding Lemma

@article{Paturi1997SatisfiabilityCL, title={Satisfiability Coding Lemma}, author={Ramamohan Paturi and Pavel Pudl{\'a}k and Francis Zane}, journal={Proceedings 38th Annual Symposium on Foundations of Computer Science}, year={1997}, pages={566-574} }

We present and analyze two simple algorithms for finding satisfying assignments of /spl kappa/-CNFs (Boolean formulae in conjunctive normal form with at most /spl kappa/ literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a satisfiable /spl kappa/-CNF formula F in time O(n/sup 2/|F|2/sup n-n//spl kappa//). The second algorithm is deterministic, and its running time approaches 2/sup n-n/2/spl kappa// for large n and…

## 207 Citations

### The complexity of unique k-SAT: an isolation lemma for k-CNFs

- Computer Science, Physics18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.
- 2003

The main technical result is an isolation lemma for k-CNFs, which shows that a given satisfiable k- CNF can be efficiently probabilistically reduced to a uniquely satisfiability k-SAT, with nontrivial, albeit exponentially small, success probability.

### Algorithms and extremal properties of SAT and CSP

- Computer Science, Mathematics
- 2011

The deterministic algorithm for satisfiability and constraint satisfaction is improved, ultimately closing the gap to Schöning’s random walk algorithm up to a subexponential factor.

### An improved exponential-time algorithm for k-SAT

- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

It is shown that, for each k, the running time of ResolveSat on a k-CNF formula is significantly better than 2/sup n/, even in the worst case, and the idea of succinctly encoding satisfying solutions can be applied to obtain lower bounds on circuit site.

### An Improved Exponential-Time Algorithm for k-SAT

- Computer ScienceFOCS
- 1998

It is shown that, for each k, the running time of ResolveSat on a k-CNF formula is significantly better than 2/sup n/, even in the worst case, and the idea of succinctly encoding satisfying solutions can be applied to obtain lower bounds on circuit site.

### Faster Random k-CNF Satisfiability

- Computer ScienceICALP
- 2020

An algorithm to solve the problem of Boolean CNF-Satisfiability when the input formula is chosen randomly by counting how many clauses are satisfied by each randomly sampled assignment, and only search in the neighborhoods of assignments with abnormally many satisfied clauses is described.

### Improved Exponential Algorithms for SAT and ClSP

- Computer Science, Mathematics
- 2015

The PPSZ algorithm is re-analyzed and it is shown that the bounds shown in the case where the input formula has at most one satisfying assignment (Unique k-SAT) hold in general, which was previously only known for k ≥ 5.

### Complexity of k-SAT

- Mathematics, Computer ScienceProceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)
- 1999

This paper shows that s/sub k/ is an increasing sequence assuming ETH for k-SAT, and shows that d>0.1/s/sub /spl infin// is the limit of s/ sub k/.

### Fighting Perebor: New and Improved Algorithms for Formula and QBF Satisfiability

- Computer Science, Mathematics2010 IEEE 51st Annual Symposium on Foundations of Computer Science
- 2010

It is shown that every function computable by linear-size formulae can be represented by decision trees of size $2^{n - \Omega(n)}$.

### A Randomized Algorithm for 3-SAT

- Mathematics, Computer ScienceMath. Comput. Sci.
- 2010

Two simple randomized algorithms PPZ and DEL are proposed for k-SAT and it is shown that when the average number of clauses for a variable that appear as unique true literals in one or more critical clauses in $$phi}$$ is between 1 and 2/(3 · log (3/2)), combined algorithm performs better than the PPZ algorithm.

### The Complexity of Satisfiability of Small Depth Circuits

- Computer ScienceIWPEC
- 2009

An improved randomized algorithm for the satisfiability problem for circuits of constant depth d and a linear number of gates cn is shown: for each d and c, the running time is 2(1 ? ?)n where the improvement $\delta\geq 1/O(c^{2-2-1}\lg^{3\cdot 2^{d-2}- 2}c)$, and the constant in the big-Oh depends only on d.

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