1 research outputs found
Backdoors to Acyclic SAT
Backdoor sets, a notion introduced by Williams et al. in 2003, are certain
sets of key variables of a CNF formula F that make it easy to solve the
formula; by assigning truth values to the variables in a backdoor set, the
formula gets reduced to one or several polynomial-time solvable formulas. More
specifically, a weak backdoor set of F is a set X of variables such that there
exits a truth assignment t to X that reduces F to a satisfiable formula F[t]
that belongs to a polynomial-time decidable base class C. A strong backdoor set
is a set X of variables such that for all assignments t to X, the reduced
formula F[t] belongs to C.
We study the problem of finding backdoor sets of size at most k with respect
to the base class of CNF formulas with acyclic incidence graphs, taking k as
the parameter. We show that
1. the detection of weak backdoor sets is W[2]-hard in general but
fixed-parameter tractable for r-CNF formulas, for any fixed r>=3, and
2. the detection of strong backdoor sets is fixed-parameter approximable.
Result 1 is the the first positive one for a base class that does not have a
characterization with obstructions of bounded size. Result 2 is the first
positive one for a base class for which strong backdoor sets are more powerful
than deletion backdoor sets.
Not only SAT, but also #SAT can be solved in polynomial time for CNF formulas
with acyclic incidence graphs. Hence Result 2 establishes a new structural
parameter that makes #SAT fixed-parameter tractable and that is incomparable
with known parameters such as treewidth and clique-width.
We obtain the algorithms by a combination of an algorithmic version of the
Erd\"os-P\'osa Theorem, Courcelle's model checking for monadic second order
logic, and new combinatorial results on how disjoint cycles can interact with
the backdoor set