Summary: Lower bounds for k­DNF Resolution on random 3­CNFs
Michael Alekhnovich #
February 24, 2005
Abstract
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.
1 Introduction
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.

Source: Alekhnovich, Michael - Institute for Advanced Study, Princeton University

Collections: Mathematics; Computer Technologies and Information Sciences