776 research outputs found
Generalizing backdoors
Abstract. A powerful intuition in the design of search methods is that one wants to proactively select variables that simplify the problem instance as much as possible when these variables are assigned values. The notion of “Backdoor ” variables follows this intuition. In this work we generalize Backdoors in such a way to allow more general classes of sub-solvers, both complete and heuristic. In order to do so, Pseudo-Backdoors and Heuristic-Backdoors are formally introduced and then applied firstly to a simple Multiple Knapsack Problem and secondly to a complex combinatorial optimization problem in the area of stochastic inventory control. Our preliminary computational experience shows the effectiveness of these approaches that are able to produce very low run times and — in the case of Heuristic-Backdoors — high quality solutions by employing very simple heuristic rules such as greedy local search strategies.
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
- …