Summary: Lower bounds for kDNF Resolution on random 3CNFs
Michael Alekhnovich #
February 24, 2005
We prove exponential lower bounds for the refutation of a random 3CNF with linear number
of clauses by kDNF Resolution for k # # log n/ log log n. For this we design a specially tailored
random restrictions that preserve the structure of the input random 3CNF while mapping every
kDNF with large covering number to 1 with high probability. Next we make use of the switching
lemma for small restrictions by Segerlind, Buss and Impagliazzo to prove the lower bound.
This work improves the previously known lower bound for Res(2) system on random 3
CNFs by Atserias, Bonet and Esteban and the result of Segerlind, Buss, Impagliazzo stating
that random O(k 2 )CNF do not possess short Res(k) refutations.
Random 3CNF formulas form the most popular stochastic distribution on the instances of NP
complete problems, the complexity of which has been investigated by many researchers. These
formulas are generated as a random subset of #n clauses over n variables, the goal is to satisfy all
the clauses. It is known that for any n there exists a sharp density threshold c n [Fri99], below which
the random formulas are almost certainly satisfiable and above which are unsatisfiable w.h.p. It is
conjectured (and there is also some empirical evidence) that solving random 3CNFs becomes very
hard when the density is close to the threshold.