TY - GEN

T1 - On smoothed k-CNF formulas and the Walksat algorithm

AU - Coja-Oghlan, Amin

AU - Feige, Uriel

AU - Frieze, Alan

AU - Krivelevich, Michael

AU - Vilenchik, Dan

PY - 2009/1/1

Y1 - 2009/1/1

N2 - In this paper we study the model of ε-smoothed k-CNF formulas. Starting from an arbitrary instance F with n variables and m = dn clauses, apply the ε-smoothing operation of flipping the polarity of every literal in every clause independently at random with probability ε. Keeping ε and k fixed, and letting the density d = m/n grow, it is rather easy to see that for d ≥ ε-k ln 2, F becomes whp unsatisfiable after smoothing. We show that a lower density that behaves roughly like ε-k+1 suffices for this purpose. We also show that our bound on d is nearly best possible in the sense that there are k-CNF formulas F of slightly lower density that whp remain satisfiable after smoothing. One consequence of our proof is a new lower bound of Ω(2k/k 2) on the density up to which Walksat solves random k-CNFs in polynomial time whp. We are not aware of any previous rigorous analysis showing that Walksat is successful at densities that are increasing as a function of k.

AB - In this paper we study the model of ε-smoothed k-CNF formulas. Starting from an arbitrary instance F with n variables and m = dn clauses, apply the ε-smoothing operation of flipping the polarity of every literal in every clause independently at random with probability ε. Keeping ε and k fixed, and letting the density d = m/n grow, it is rather easy to see that for d ≥ ε-k ln 2, F becomes whp unsatisfiable after smoothing. We show that a lower density that behaves roughly like ε-k+1 suffices for this purpose. We also show that our bound on d is nearly best possible in the sense that there are k-CNF formulas F of slightly lower density that whp remain satisfiable after smoothing. One consequence of our proof is a new lower bound of Ω(2k/k 2) on the density up to which Walksat solves random k-CNFs in polynomial time whp. We are not aware of any previous rigorous analysis showing that Walksat is successful at densities that are increasing as a function of k.

UR - http://www.scopus.com/inward/record.url?scp=70349134696&partnerID=8YFLogxK

U2 - 10.1137/1.9781611973068.50

DO - 10.1137/1.9781611973068.50

M3 - Conference contribution

AN - SCOPUS:70349134696

SN - 9780898716801

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 451

EP - 460

BT - Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms

PB - Association for Computing Machinery

T2 - 20th Annual ACM-SIAM Symposium on Discrete Algorithms

Y2 - 4 January 2009 through 6 January 2009

ER -