80 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
Upper and Lower Bounds for Weak Backdoor Set Detection
We obtain upper and lower bounds for running times of exponential time
algorithms for the detection of weak backdoor sets of 3CNF formulas,
considering various base classes. These results include (omitting polynomial
factors), (i) a 4.54^k algorithm to detect whether there is a weak backdoor set
of at most k variables into the class of Horn formulas; (ii) a 2.27^k algorithm
to detect whether there is a weak backdoor set of at most k variables into the
class of Krom formulas. These bounds improve an earlier known bound of 6^k. We
also prove a 2^k lower bound for these problems, subject to the Strong
Exponential Time Hypothesis.Comment: A short version will appear in the proceedings of the 16th
International Conference on Theory and Applications of Satisfiability Testin
Backdoors to Normality for Disjunctive Logic Programs
Over the last two decades, propositional satisfiability (SAT) has become one
of the most successful and widely applied techniques for the solution of
NP-complete problems. The aim of this paper is to investigate theoretically how
Sat can be utilized for the efficient solution of problems that are harder than
NP or co-NP. In particular, we consider the fundamental reasoning problems in
propositional disjunctive answer set programming (ASP), Brave Reasoning and
Skeptical Reasoning, which ask whether a given atom is contained in at least
one or in all answer sets, respectively. Both problems are located at the
second level of the Polynomial Hierarchy and thus assumed to be harder than NP
or co-NP. One cannot transform these two reasoning problems into SAT in
polynomial time, unless the Polynomial Hierarchy collapses. We show that
certain structural aspects of disjunctive logic programs can be utilized to
break through this complexity barrier, using new techniques from Parameterized
Complexity. In particular, we exhibit transformations from Brave and Skeptical
Reasoning to SAT that run in time O(2^k n^2) where k is a structural parameter
of the instance and n the input size. In other words, the reduction is
fixed-parameter tractable for parameter k. As the parameter k we take the size
of a smallest backdoor with respect to the class of normal (i.e.,
disjunction-free) programs. Such a backdoor is a set of atoms that when deleted
makes the program normal. In consequence, the combinatorial explosion, which is
expected when transforming a problem from the second level of the Polynomial
Hierarchy to the first level, can now be confined to the parameter k, while the
running time of the reduction is polynomial in the input size n, where the
order of the polynomial is independent of k.Comment: A short version will appear in the Proceedings of the Proceedings of
the 27th AAAI Conference on Artificial Intelligence (AAAI'13). A preliminary
version of the paper was presented on the workshop Answer Set Programming and
Other Computing Paradigms (ASPOCP 2012), 5th International Workshop,
September 4, 2012, Budapest, Hungar
Limits of Preprocessing
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a
complexity theoretic assumption, none of the considered problems can be reduced
by polynomial-time preprocessing to a problem kernel whose size is polynomial
in a structural problem parameter of the input, such as induced width or
backdoor size. Our results provide a firm theoretical boundary for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: This is a slightly longer version of a paper that appeared in the
proceedings of AAAI 201
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