552 research outputs found
A Logical Approach to Efficient Max-SAT solving
Weighted Max-SAT is the optimization version of SAT and many important
problems can be naturally encoded as such. Solving weighted Max-SAT is an
important problem from both a theoretical and a practical point of view. In
recent years, there has been considerable interest in finding efficient solving
techniques. Most of this work focus on the computation of good quality lower
bounds to be used within a branch and bound DPLL-like algorithm. Most often,
these lower bounds are described in a procedural way. Because of that, it is
difficult to realize the {\em logic} that is behind.
In this paper we introduce an original framework for Max-SAT that stresses
the parallelism with classical SAT. Then, we extend the two basic SAT solving
techniques: {\em search} and {\em inference}. We show that many algorithmic
{\em tricks} used in state-of-the-art Max-SAT solvers are easily expressable in
{\em logic} terms with our framework in a unified manner.
Besides, we introduce an original search algorithm that performs a restricted
amount of {\em weighted resolution} at each visited node. We empirically
compare our algorithm with a variety of solving alternatives on several
benchmarks. Our experiments, which constitute to the best of our knowledge the
most comprehensive Max-sat evaluation ever reported, show that our algorithm is
generally orders of magnitude faster than any competitor
Solving MaxSAT and #SAT on structured CNF formulas
In this paper we propose a structural parameter of CNF formulas and use it to
identify instances of weighted MaxSAT and #SAT that can be solved in polynomial
time. Given a CNF formula we say that a set of clauses is precisely satisfiable
if there is some complete assignment satisfying these clauses only. Let the
ps-value of the formula be the number of precisely satisfiable sets of clauses.
Applying the notion of branch decompositions to CNF formulas and using ps-value
as cut function, we define the ps-width of a formula. For a formula given with
a decomposition of polynomial ps-width we show dynamic programming algorithms
solving weighted MaxSAT and #SAT in polynomial time. Combining with results of
'Belmonte and Vatshelle, Graph classes with structured neighborhoods and
algorithmic applications, Theor. Comput. Sci. 511: 54-65 (2013)' we get
polynomial-time algorithms solving weighted MaxSAT and #SAT for some classes of
structured CNF formulas. For example, we get algorithms for
formulas of clauses and variables and size , if has a linear
ordering of the variables and clauses such that for any variable occurring
in clause , if appears before then any variable between them also
occurs in , and if appears before then occurs also in any clause
between them. Note that the class of incidence graphs of such formulas do not
have bounded clique-width
Certifying Correctness for Combinatorial Algorithms : by Using Pseudo-Boolean Reasoning
Over the last decades, dramatic improvements in combinatorialoptimisation algorithms have significantly impacted artificialintelligence, operations research, and other areas. These advances,however, are achieved through highly sophisticated algorithms that aredifficult to verify and prone to implementation errors that can causeincorrect results. A promising approach to detect wrong results is touse certifying algorithms that produce not only the desired output butalso a certificate or proof of correctness of the output. An externaltool can then verify the proof to determine that the given answer isvalid. In the Boolean satisfiability (SAT) community, this concept iswell established in the form of proof logging, which has become thestandard solution for generating trustworthy outputs. The problem isthat there are still some SAT solving techniques for which prooflogging is challenging and not yet used in practice. Additionally,there are many formalisms more expressive than SAT, such as constraintprogramming, various graph problems and maximum satisfiability(MaxSAT), for which efficient proof logging is out of reach forstate-of-the-art techniques.This work develops a new proof system building on the cutting planesproof system and operating on pseudo-Boolean constraints (0-1 linearinequalities). We explain how such machine-verifiable proofs can becreated for various problems, including parity reasoning, symmetry anddominance breaking, constraint programming, subgraph isomorphism andmaximum common subgraph problems, and pseudo-Boolean problems. Weimplement and evaluate the resulting algorithms and a verifier for theproof format, demonstrating that the approach is practical for a widerange of problems. We are optimistic that the proposed proof system issuitable for designing certifying variants of algorithms inpseudo-Boolean optimisation, MaxSAT and beyond
On complexity of optimized crossover for binary representations
We consider the computational complexity of producing the best possible
offspring in a crossover, given two solutions of the parents. The crossover
operators are studied on the class of Boolean linear programming problems,
where the Boolean vector of variables is used as the solution representation.
By means of efficient reductions of the optimized gene transmitting crossover
problems (OGTC) we show the polynomial solvability of the OGTC for the maximum
weight set packing problem, the minimum weight set partition problem and for
one of the versions of the simple plant location problem. We study a connection
between the OGTC for linear Boolean programming problem and the maximum weight
independent set problem on 2-colorable hypergraph and prove the NP-hardness of
several special cases of the OGTC problem in Boolean linear programming.Comment: Dagstuhl Seminar 06061 "Theory of Evolutionary Algorithms", 200
Symmetry-breaking Answer Set Solving
In the context of Answer Set Programming, this paper investigates
symmetry-breaking to eliminate symmetric parts of the search space and,
thereby, simplify the solution process. We propose a reduction of disjunctive
logic programs to a coloured digraph such that permutational symmetries can be
constructed from graph automorphisms. Symmetries are then broken by introducing
symmetry-breaking constraints. For this purpose, we formulate a preprocessor
that integrates a graph automorphism system. Experiments demonstrate its
computational impact.Comment: Proceedings of ICLP'10 Workshop on Answer Set Programming and Other
Computing Paradig
Quantum Algorithm for Variant Maximum Satisfiability
In this paper, we proposed a novel quantum algorithm for the maximum satisfiability problem. Satisfiability (SAT) is to find the set of assignment values of input variables for the given Boolean function that evaluates this function as TRUE or prove that such satisfying values do not exist. For a POS SAT problem, we proposed a novel quantum algorithm for the maximum satisfiability (MAX-SAT), which returns the maximum number of OR terms that are satisfied for the SAT-unsatisfiable function, providing us with information on how far the given Boolean function is from the SAT satisfaction. We used Groverâs algorithm with a new block called quantum counter in the oracle circuit. The proposed circuit can be adapted for various forms of satisfiability expressions and several satisfiability-like problems. Using the quantum counter and mirrors for SAT terms reduces the need for ancilla qubits and realizes a large Toffoli gate that is then not needed. Our circuit reduces the number of ancilla qubits for the terms T of the Boolean function from T of ancilla qubits to ââlog2âĄTâ+1. We analyzed and compared the quantum cost of the traditional oracle design with our design which gives a low quantum cost
Parameterized Complexity of Graph Constraint Logic
Graph constraint logic is a framework introduced by Hearn and Demaine, which
provides several problems that are often a convenient starting point for
reductions. We study the parameterized complexity of Constraint Graph
Satisfiability and both bounded and unbounded versions of Nondeterministic
Constraint Logic (NCL) with respect to solution length, treewidth and maximum
degree of the underlying constraint graph as parameters. As a main result we
show that restricted NCL remains PSPACE-complete on graphs of bounded
bandwidth, strengthening Hearn and Demaine's framework. This allows us to
improve upon existing results obtained by reduction from NCL. We show that
reconfiguration versions of several classical graph problems (including
independent set, feedback vertex set and dominating set) are PSPACE-complete on
planar graphs of bounded bandwidth and that Rush Hour, generalized to boards, is PSPACE-complete even when is at most a constant
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