Tighter Hard Instances for PPSZ

@article{Pudlk2017TighterHI,
  title={Tighter Hard Instances for PPSZ},
  author={Pavel Pudl{\'a}k and Dominik Scheder and Navid Talebanfard},
  journal={ArXiv},
  year={2017},
  volume={abs/1611.01291}
}
We construct uniquely satisfiable k-CNF formulas that are hard for the PPSZ algorithm, the currently best known algorithm solving k-SAT. This algorithm tries to generate a satisfying assignment by picking a random variable at a time and attempting to derive its value using some inference heuristic and otherwise assigning a random value. The "weak PPSZ" checks all subformulas of a given size to derive a value and the "strong PPSZ" runs resolution with width bounded by some given function… 
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References

SHOWING 1-10 OF 20 REFERENCES
An improved exponential-time algorithm for k-SAT
TLDR
A simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form, which is the fastest known probabilistic algorithm for k-CNF satisfiability and proves a lower bound on the number of codewords of a code defined by a <i>k</i>-C NF.
Exponential Lower Bounds for the PPSZ k-SAT Algorithm
TLDR
This paper constructs hard instances for PPSZ, an elegant randomized algorithm for k-SAT that proves that its expected running time on k-CNF formulas with n variables is at most 2(1−ek)n, where ek e Ω(1/k).
3-SAT Faster and Simpler - Unique-SAT Bounds for PPSZ Hold in General
TLDR
The PPSZ algorithm by Paturi, Pudlak, Saks, and Zane is shown to be the fastest known algorithm for Unique $k$-SAT, where the input formula does not have more than one satisfying assignment, and it is shown that this is also the case for k=3,4.
3-SAT Faster and Simpler - Unique-SAT Bounds for PPSZ Hold in General
  • Timon Hertli
  • Mathematics, Computer Science
    2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
  • 2011
TLDR
The PPSZ algorithm by Paturi, Pudl\'ak, Saks, and Zane is shown to be the fastest known algorithm for Unique k-SAT, where the input formula does not have more than one satisfying assignment, and it is shown that this is also the case for k=3,4.
Complexity of kSAT
TLDR
The complexity of k-SAT is examined, and a relationship that governs the complexity ofk-S AT for various cases is derived under the assumption that k- SAT does not have subexponential algorithms for k 3.
A deterministic (2-2/(k+1))n algorithm for k-SAT based on local search
Edit Distance Cannot Be Computed in Strongly Subquadratic Time (unless SETH is false)
TLDR
This paper shows that, if the edit distance can be computed in time O(n2-δ) for some constant δ>0, then the satisfiability of conjunctive normal form formulas with N variables and M clauses can be solved in time MO(1) 2(1-ε)N for a constant ε>0.
Satisfiability Coding Lemma
TLDR
This basic lemma shows how to encode satisfying solutions of a /spl kappa/-CNF succinctly as well as an upper and lower bound on the size of depth 3 circuits of AND and OR gates computing the parity function.
Random Cnf’s are Hard for the Polynomial Calculus
TLDR
The equivalence of refutation-degree and Gaussian width allows us to also simplify the refutations-degree lower bounds of Buss et al. (2001) and additionally prove non-trivial upper bounds on the resolution and PC complexity of refuting unsatisfiable systems of linear equations.
Complexity of k-SAT
  • R. Impagliazzo, R. Paturi
  • Mathematics, Computer Science
    Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)
  • 1999
TLDR
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/.
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