24,349 research outputs found
Parameterized complexity of DPLL search procedures
We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
- …