 
Summary: SIAM J. COMPUT. c 2006 Society for Industrial and Applied Mathematics
Vol. 36, No. 5, pp. 12481263
LINEAR UPPER BOUNDS FOR RANDOM WALK ON SMALL
DENSITY RANDOM 3CNFs
MIKHAIL ALEKHNOVICH AND ELI BENSASSON
In memory of Mikhail (Misha) Alekhnovichfriend, colleague and brilliant mind
Abstract. We analyze the efficiency of the random walk algorithm on random 3CNF instances
and prove linear upper bounds on the running time of this algorithm for small clause density, less
than 1.63. This is the first subexponential upper bound on the running time of a local improvement
algorithm on random instances. Our proof introduces a simple, yet powerful tool for analyzing such
algorithms, which may be of further use. This object, called a terminator, is a weighted satisfying
assignment. We show that any CNF having a good (small weight) terminator is assured to be solved
quickly by the random walk algorithm. This raises the natural question of the terminator threshold
which is the maximal clause density for which such assignments exist (with high probability). We
use the analysis of the pure literal heuristic presented by Broder, Frieze, and Upfal [Proceedings of
the Fourth Annual ACMSIAM Symposium on Discrete Algorithms, 1993, pp. 322330] and Luby,
Mitzenmacher, and Shokrollahi [Proceedings of the Ninth Annual ACMSIAM Symposium on Dis
crete Algorithms, 1998, pp. 364373] and show that for small clause densities good terminators exist.
Thus we show that the pure literal threshold (1.63) is a lower bound on the terminator threshold.
(We conjecture the terminator threshold to be in fact higher.) One nice property of terminators is
