14,104 research outputs found
Effective problem solving using SAT solvers
In this article we demonstrate how to solve a variety of problems and puzzles
using the built-in SAT solver of the computer algebra system Maple. Once the
problems have been encoded into Boolean logic, solutions can be found (or shown
to not exist) automatically, without the need to implement any search
algorithm. In particular, we describe how to solve the -queens problem, how
to generate and solve Sudoku puzzles, how to solve logic puzzles like the
Einstein riddle, how to solve the 15-puzzle, how to solve the maximum clique
problem, and finding Graeco-Latin squares.Comment: To appear in Proceedings of the Maple Conference 201
Automatically improving SAT encoding of constraint problems through common subexpression elimination in Savile Row
The formulation of a Propositional Satisfiability (SAT) problem instance is vital to efficient solving. This has motivated research on preprocessing, and inprocessing techniques where reformulation of a SAT instance is interleaved with solving. Preprocessing and inprocessing are highly effective in extending the reach of SAT solvers, however they necessarily operate on the lowest level representation of the problem, the raw SAT clauses, where higher-level patterns are difficult and/or costly to identify. Our approach is different: rather than reformulate the SAT representation directly, we apply automated reformulations to a higher level representation (a constraint model) of the original problem. Common Subexpression Elimination (CSE) is a family of techniques to improve automatically the formulation of constraint satisfaction problems, which are often highly beneficial when using a conventional constraint solver. In this work we demonstrate that CSE has similar benefits when the reformulated constraint model is encoded to SAT and solved using a state-of-the-art SAT solver. In some cases we observe speed improvements of over 100 times
On CNF Conversion for SAT Enumeration
Modern SAT solvers are designed to handle problems expressed in Conjunctive
Normal Form (CNF) so that non-CNF problems must be CNF-ized upfront, typically
by using variants of either Tseitin or Plaisted&Greenbaum transformations. When
passing from solving to enumeration, however, the capability of producing
partial satisfying assignment that are as small as possible becomes crucial,
which raises the question of whether such CNF encodings are also effective for
enumeration. In this paper, we investigate both theoretically and empirically
the effectiveness of CNF conversions for SAT enumeration. On the negative side,
we show that: (i) Tseitin transformation prevents the solver from producing
short partial assignments, thus seriously affecting the effectiveness of
enumeration; (ii) Plaisted&Greenbaum transformation overcomes this problem only
in part. On the positive side, we show that combining Plaisted&Greenbaum
transformation with NNF preprocessing upfront -- which is typically not used in
solving -- can fully overcome the problem and can drastically reduce both the
number of partial assignments and the execution time.Comment: 14 pages, 12 figure
On CNF Conversion for Disjoint SAT Enumeration
Modern SAT solvers are designed to handle problems expressed in Conjunctive Normal Form (CNF) so that non-CNF problems must be CNF-ized upfront, typically by using variants of either Tseitin or Plaisted and Greenbaum transformations. When passing from solving to enumeration, however, the capability of producing partial satisfying assignments that are as small as possible becomes crucial, which raises the question of whether such CNF encodings are also effective for enumeration.
In this paper, we investigate both theoretically and empirically the effectiveness of CNF conversions for disjoint SAT enumeration. On the negative side, we show that: (i) Tseitin transformation prevents the solver from producing short partial assignments, thus seriously affecting the effectiveness of enumeration; (ii) Plaisted and Greenbaum transformation overcomes this problem only in part. On the positive side, we show that combining Plaisted and Greenbaum transformation with NNF preprocessing upfront - which is typically not used in solving - can fully overcome the problem and can drastically reduce both the number of partial assignments and the execution time
An Iterative Path-Breaking Approach with Mutation and Restart Strategies for the MAX-SAT Problem
Although Path-Relinking is an effective local search method for many
combinatorial optimization problems, its application is not straightforward in
solving the MAX-SAT, an optimization variant of the satisfiability problem
(SAT) that has many real-world applications and has gained more and more
attention in academy and industry. Indeed, it was not used in any recent
competitive MAX-SAT algorithms in our knowledge. In this paper, we propose a
new local search algorithm called IPBMR for the MAX-SAT, that remedies the
drawbacks of the Path-Relinking method by using a careful combination of three
components: a new strategy named Path-Breaking to avoid unpromising regions of
the search space when generating trajectories between two elite solutions; a
weak and a strong mutation strategies, together with restarts, to diversify the
search; and stochastic path generating steps to avoid premature local optimum
solutions. We then present experimental results to show that IPBMR outperforms
two of the best state-of-the-art MAX-SAT solvers, and an empirical
investigation to identify and explain the effect of the three components in
IPBMR
Breaking Instance-Independent Symmetries In Exact Graph Coloring
Code optimization and high level synthesis can be posed as constraint
satisfaction and optimization problems, such as graph coloring used in register
allocation. Graph coloring is also used to model more traditional CSPs relevant
to AI, such as planning, time-tabling and scheduling. Provably optimal
solutions may be desirable for commercial and defense applications.
Additionally, for applications such as register allocation and code
optimization, naturally-occurring instances of graph coloring are often small
and can be solved optimally. A recent wave of improvements in algorithms for
Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests
generic problem-reduction methods, rather than problem-specific heuristics,
because (1) heuristics may be upset by new constraints, (2) heuristics tend to
ignore structure, and (3) many relevant problems are provably inapproximable.
Problem reductions often lead to highly symmetric SAT instances, and
symmetries are known to slow down SAT solvers. In this work, we compare several
avenues for symmetry breaking, in particular when certain kinds of symmetry are
present in all generated instances. Our focus on reducing CSPs to SAT allows us
to leverage recent dramatic improvement in SAT solvers and automatically
benefit from future progress. We can use a variety of black-box SAT solvers
without modifying their source code because our symmetry-breaking techniques
are static, i.e., we detect symmetries and add symmetry breaking predicates
(SBPs) during pre-processing.
An important result of our work is that among the types of
instance-independent SBPs we studied and their combinations, the simplest and
least complete constructions are the most effective. Our experiments also
clearly indicate that instance-independent symmetries should mostly be
processed together with instance-specific symmetries rather than at the
specification level, contrary to what has been suggested in the literature
On Tackling the Limits of Resolution in SAT Solving
The practical success of Boolean Satisfiability (SAT) solvers stems from the
CDCL (Conflict-Driven Clause Learning) approach to SAT solving. However, from a
propositional proof complexity perspective, CDCL is no more powerful than the
resolution proof system, for which many hard examples exist. This paper
proposes a new problem transformation, which enables reducing the decision
problem for formulas in conjunctive normal form (CNF) to the problem of solving
maximum satisfiability over Horn formulas. Given the new transformation, the
paper proves a polynomial bound on the number of MaxSAT resolution steps for
pigeonhole formulas. This result is in clear contrast with earlier results on
the length of proofs of MaxSAT resolution for pigeonhole formulas. The paper
also establishes the same polynomial bound in the case of modern core-guided
MaxSAT solvers. Experimental results, obtained on CNF formulas known to be hard
for CDCL SAT solvers, show that these can be efficiently solved with modern
MaxSAT solvers
- …