The Pseudo-Boolean Optimization (PBO) and Maximum Satisfiability (MaxSAT) problems are natural optimization extensions of Boolean Satisfiability (SAT). In the recent past, different algorithms have been proposed for PBO and for MaxSAT, despite the existence of straightforward mappings from PBO to MaxSAT and vice-versa. This papers proposes Weighted Boolean Optimization (WBO), a new unified framework that aggregates and extends PBO and MaxSAT. In addition, the paper proposes a new unsatisfiability-based algorithm for WBO, based on recent unsatisfiability-based algorithms for MaxSAT. Besides standard MaxSAT, the new algorithm can also be used to solve weighted MaxSAT and PBO, handling pseudo-Boolean constraints either natively or by translation to clausal form. Experimental results illustrate that unsatisfiability-based algorithms for MaxSAT can be orders of magnitude more efficient than existing dedicated algorithms. Finally, the paper illustrates how other algorithms for either PBO or MaxSAT can be extended to WBO.
Deep Dive into Algorithms for Weighted Boolean Optimization.
The Pseudo-Boolean Optimization (PBO) and Maximum Satisfiability (MaxSAT) problems are natural optimization extensions of Boolean Satisfiability (SAT). In the recent past, different algorithms have been proposed for PBO and for MaxSAT, despite the existence of straightforward mappings from PBO to MaxSAT and vice-versa. This papers proposes Weighted Boolean Optimization (WBO), a new unified framework that aggregates and extends PBO and MaxSAT. In addition, the paper proposes a new unsatisfiability-based algorithm for WBO, based on recent unsatisfiability-based algorithms for MaxSAT. Besides standard MaxSAT, the new algorithm can also be used to solve weighted MaxSAT and PBO, handling pseudo-Boolean constraints either natively or by translation to clausal form. Experimental results illustrate that unsatisfiability-based algorithms for MaxSAT can be orders of magnitude more efficient than existing dedicated algorithms. Finally, the paper illustrates how other algorithms for either PBO o
In the area of Boolean-based decision and optimization procedures, natural extensions of Boolean Satisfiability (SAT) include Maximum Satisfiability (MaxSAT) [10] and Pseudo-Boolean Optimization (PBO) [6]. Algorithms for MaxSAT and PBO have been the subject of significant improvements over the last few years. This in turn, motivated the use of both PBO and, more recently, of MaxSAT in a number of practical applications. Interestingly, albeit there are simple translations from any MaxSAT variant to PBO and vice-versa (by encoding to CNF) [1,18], algorithms for MaxSAT and PBO have evolved separately, and often use fairly different algorithmic organizations. Nevertheless, there exists work that acknowledges this relationship and algorithms that can solve instances of MaxSAT and of PBO have already been proposed [1,18].
Recent work has provided more alternatives for solving either MaxSAT or PBO, by using SAT solvers and the identification of unsatisfiable sub-formulas [16,27]. However, the proposed algorithms were restricted to the plain and partial variants of MaxSAT and to a restricted form of Binate Covering for PBO. This paper extends this recent work in a number of directions. First, the paper proposes a simple algorithm for (Partial) Weighted MaxSAT, using unsatisfiable sub-formula identification. Second, the paper generalizes MaxSAT and PBO by introducing Weighted Boolean Optimization (WBO), a new modeling framework for solving linear optimization problems over Boolean domains. Third, the paper shows how to extend the unsatisfiability-based algorithm for MaxSAT for solving WBO problems. Finally, the paper suggests how other algorithms can be used for solving WBO. Besides the proposed contributions, the paper also provides empirical evidence that unsatisfiability-based MaxSAT and WBO solvers can outperform state-of-the-art solvers on problem instances from practical problems.
The paper is organized as follows. Section 2 provides a brief overview of the topics addressed in the paper, namely MaxSAT, PBO, translations from MaxSAT to PBO and vice-versa, and unsatisfiability-based algorithms for MaxSAT. Section 3 details an algorithm for (Partial) Weighted MaxSAT based on unsatisfiable sub-formula identification. Next, Section 4 introduces Weighted Boolean Optimization (WBO), and shows how to extend the algorithm of Section 3 to WBO. Section 5 analyzes the experimental results, obtained on representative classes of problem instances. Section 6 overviews related work, and Section 7 concludes the paper.
This section briefly introduces the Maximum Satisfiability (MaxSAT) problem and its variants, as well as the Pseudo-Boolean Optimization (PBO) problem. The main approaches used by state-of-the-art solvers are summarized. Moreover, translation procedures from MaxSAT to PBO and vice-versa are overviewed. Finally, unsatisfiabilitybased MaxSAT algorithms are surveyed, all of which the paper uses in later sections.
Given a CNF formula ϕ, the Maximum Satisfiability (MaxSAT) problem can be defined as finding an assignment that maximizes the number of satisfied clauses (which implies that the assignment minimizes the number of unsatisfied clauses). Besides the classical MaxSAT problem, there are also three well-known variants of MaxSAT: weighted MaxSAT, partial MaxSAT and weighted partial MaxSAT. All these formulations have been used in a wide range of practical applications, namely scheduling, FPGA routing [34], design automation [31], among others.
A partial CNF formula is described as the conjunction of two CNF formulas ϕ h and ϕ s , where ϕ h represents the hard clauses and ϕ s represents the soft clauses. The partial MaxSAT problem consists in finding an assignment to the problem variables such that all hard clauses (ϕ h ) are satisfied, and the number of satisfied soft clauses (ϕ s ) is maximixed.
A weighted CNF formula is a set of weighted clauses. A weighted clause is a pair (ω, c), where ω is a classical clause and c is a natural number corresponding to the cost of unsatisfying ω. Given a weighted CNF formula, the weighted MaxSAT problem consists in finding an assignment to the problem variables such that the total weight of the satified clauses is maximized (which implies that the total weight of the unsatisfied clauses is minimized).
A weighted partial CNF formula is the conjunction of a weighted CNF formula (soft clauses) and a classical CNF formula (hard clauses). The weighted partial MaxSAT problem consists in finding an assignment to the variables such that all hard clauses are satisfied and the total weight of satisfied soft clauses is maximized. Observe that, for both partial MaxSAT and weighted partial MaxSAT, hard clauses can also be represented as weighted clauses: one can consider that the weight is greater than the sum of the weights of the soft clauses.
Starting with the seminal work of Borchers and Furman [10], there has been an increasing interest in developing efficient MaxSAT solvers. F
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