654 research outputs found
Incremental Cardinality Constraints for MaxSAT
Maximum Satisfiability (MaxSAT) is an optimization variant of the Boolean
Satisfiability (SAT) problem. In general, MaxSAT algorithms perform a
succession of SAT solver calls to reach an optimum solution making extensive
use of cardinality constraints. Many of these algorithms are non-incremental in
nature, i.e. at each iteration the formula is rebuilt and no knowledge is
reused from one iteration to another. In this paper, we exploit the knowledge
acquired across iterations using novel schemes to use cardinality constraints
in an incremental fashion. We integrate these schemes with several MaxSAT
algorithms. Our experimental results show a significant performance boost for
these algo- rithms as compared to their non-incremental counterparts. These
results suggest that incremental cardinality constraints could be beneficial
for other constraint solving domains.Comment: 18 pages, 4 figures, 1 table. Final version published in Principles
and Practice of Constraint Programming (CP) 201
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
A hybrid constraint programming and semidefinite programming approach for the stable set problem
This work presents a hybrid approach to solve the maximum stable set problem,
using constraint and semidefinite programming. The approach consists of two
steps: subproblem generation and subproblem solution. First we rank the
variable domain values, based on the solution of a semidefinite relaxation.
Using this ranking, we generate the most promising subproblems first, by
exploring a search tree using a limited discrepancy strategy. Then the
subproblems are being solved using a constraint programming solver. To
strengthen the semidefinite relaxation, we propose to infer additional
constraints from the discrepancy structure. Computational results show that the
semidefinite relaxation is very informative, since solutions of good quality
are found in the first subproblems, or optimality is proven immediately.Comment: 14 page
Statistical mechanics of budget-constrained auctions
Finding the optimal assignment in budget-constrained auctions is a
combinatorial optimization problem with many important applications, a notable
example being the sale of advertisement space by search engines (in this
context the problem is often referred to as the off-line AdWords problem).
Based on the cavity method of statistical mechanics, we introduce a message
passing algorithm that is capable of solving efficiently random instances of
the problem extracted from a natural distribution, and we derive from its
properties the phase diagram of the problem. As the control parameter (average
value of the budgets) is varied, we find two phase transitions delimiting a
region in which long-range correlations arise.Comment: Minor revisio
Refined Core Relaxations for Core-Guided Maximum Satisfiability Algorithms
The so-called declarative approach has proven to be a viable paradigm for solving various real-world NP-hard optimization problems in practice. In the declarative approach, the problem at hand is encoded using a mathematical constraint language, and an algorithm for the specific language is employed to obtain optimal solutions to an instance of the problem. One of the most viable declarative optimization paradigms of the last years is maximum satisfiability (MaxSAT) with propositional logic as the constraint language.
So-called core-guided MaxSAT algorithms are arguably one of the most effective MaxSAT-solving paradigms in practice today. Core-guided algorithms iteratively detect and rule out (relax) sources of inconsistencies (so-called unsatisfiable cores) in the instance being solved. Especially effective are recent algorithmic variants of the core-guided approach which employ so-called soft cardinality constraints for ruling out inconsistencies.
In this thesis, we present a structure-sharing technique for the cardinality-based core relaxation steps performed by core-guided MaxSAT solvers. The technique aims at reducing the inherent growth in the size of the propositional formula resulting from the core relaxation steps. Additionally, it enables more efficient reasoning over the relationships between different cores.
We empirically evaluate the proposed technique on two different core-guided algorithms and provide open-source implementations of our solvers employing the technique. Our results show that the proposed structure-sharing can improve the performance of the algorithms both in theory and in practice
Optimization Modulo Theories with Linear Rational Costs
In the contexts of automated reasoning (AR) and formal verification (FV),
important decision problems are effectively encoded into Satisfiability Modulo
Theories (SMT). In the last decade efficient SMT solvers have been developed
for several theories of practical interest (e.g., linear arithmetic, arrays,
bit-vectors). Surprisingly, little work has been done to extend SMT to deal
with optimization problems; in particular, we are not aware of any previous
work on SMT solvers able to produce solutions which minimize cost functions
over arithmetical variables. This is unfortunate, since some problems of
interest require this functionality.
In the work described in this paper we start filling this gap. We present and
discuss two general procedures for leveraging SMT to handle the minimization of
linear rational cost functions, combining SMT with standard minimization
techniques. We have implemented the procedures within the MathSAT SMT solver.
Due to the absence of competitors in the AR, FV and SMT domains, we have
experimentally evaluated our implementation against state-of-the-art tools for
the domain of linear generalized disjunctive programming (LGDP), which is
closest in spirit to our domain, on sets of problems which have been previously
proposed as benchmarks for the latter tools. The results show that our tool is
very competitive with, and often outperforms, these tools on these problems,
clearly demonstrating the potential of the approach.Comment: Submitted on january 2014 to ACM Transactions on Computational Logic,
currently under revision. arXiv admin note: text overlap with arXiv:1202.140
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