## A Discrete Lagrangian-Based Global-Search Method for Solving Satisfiability Problems (1998)

Venue: | Journal of Global Optimization |

Citations: | 59 - 7 self |

### BibTeX

@ARTICLE{Wah98adiscrete,

author = {Benjamin W. Wah and Yi Shang},

title = {A Discrete Lagrangian-Based Global-Search Method for Solving Satisfiability Problems},

journal = {Journal of Global Optimization},

year = {1998},

volume = {12},

pages = {61--99}

}

### Years of Citing Articles

### OpenURL

### Abstract

Satisfiability is a class of NP-complete problems that model a wide range of real-world applications. These problems are difficult to solve because they have many local minima in their search space, often trapping greedy search methods that utilize some form of descent. In this paper, we propose a new discrete Lagrange-multiplier-based global-search method for solving satisfiability problems. We derive new approaches for applying Lagrangian methods in discrete space, show that equilibrium is reached when a feasible assignment to the original problem is found, and present heuristic algorithms to look for equilibrium points. Instead of restarting from a new starting point when a search reaches a local trap, the Lagrange multipliers in our method provide a force to lead the search out of a local minimum and move it in the direction provided by the Lagrange multipliers. One of the major advantages of our method is that it has very few algorithmic parameters to be tuned by users, and the se...

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Citation Context ...s of Grasp on the DIMACS benchmark problems [39]. Grasp is a greedy randomized adaptive search procedure that can find good quality solutions for a wide variety of combinatorial optimization problems =-=[7, 32, 8, 40]. In [39], four implementations of Grasp were applied to solve fi-=-ve classes of DIMACS SAT problems, "aim," "ii," "jnh," "ssa7552," and "par." Comparing to GSAT, Grasp did better on the "aim," "ssa7552... |

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Citation Context ...s of Grasp on the DIMACS benchmark problems [39]. Grasp is a greedy randomized adaptive search procedure that can find good quality solutions for a wide variety of combinatorial optimization problems =-=[7, 32, 8, 40]. In [39], four implementations of Grasp were applied to solve fi-=-ve classes of DIMACS SAT problems, "aim," "ii," "jnh," "ssa7552," and "par." Comparing to GSAT, Grasp did better on the "aim," "ssa7552... |

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Citation Context ...f A 3 on some "g" problems. Recall that A 3 was developed to cope with large flat plateaus in the search space that confuse A 2 , which failed to find any solution within 5 million iteration=-=s. Hansen [HJ90]-=- and later Selman [SKC93] addressed this problem by using the tabu search strategy. In a similar way, we have adopted this strategy in A 3 by keeping a tabu list to prevent flipping the same variable ... |

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Citation Context ...n (1.1). Methods to solve it can be either complete or incomplete, depending on their ability to prove infeasibility. Complete methods for solving (1.1) include resolution [Rob65, GN87], backtracking =-=[Pur83]-=- and consistency testings[Gu89, GW92, Guar]. An important resolution method is Davis-Putnam's algorithm [DP60]. These methods enumerative the search space systematically, and may rely on incomplete me... |

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Citation Context ...s are very close to 1 and, therefore, overlap with the curve showing the minimum Lagrange multiplier values. See Figure 5 for further explanation. ffl Circuit synthesis problems (ii) by Kamath et al. =-=[30]-=- --- a set of SAT encodings of Boolean circuit-synthesis problems; ffl Circuit diagnosis problems (ssa) --- a set of SAT formulas based on circuit fault analysis; ffl Parity learning problems (par) --... |

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Citation Context ...ue may be large. For instance, in solving MAX-SAT problems, the objective representing the weighted sum of the number of unsatisfied clauses can be large and can provide better guidance in the search =-=[49]-=-. This part of the search is similar to what is done in many local search methods, such as Gu's local-search methods [18, 19, 20] and GSAT [48, 44, 45, 47, 42, 46], which descends into local minima in... |

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Citation Context ...). Pure descent methods are not suitable when there are constraints in the search space as formulated in (2). Recently, some local search methods were proposed and applied to solve large SAT problems =-=[37, 11, 5, 39]-=-. The most notable ones are those developed independently by Gu and Selman. Gu developed a group of local search methods for solving SAT and CSP problems. In his Ph.D thesis [14], he first formulated ... |

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Citation Context ...lems [Mor93, GW93, DTWZ94]. The most notable ones are those developed independently by Gu and Selman. Gu developed a group of local search methods for solving SAT and CSP problems. In his Ph.D thesis =-=[Gu89]-=-, he first formulated conflicts in the objective function and proposed a discrete relaxation algorithm (a class of deterministic local search) to minimize the number of conflicts in these problems. Th... |

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Citation Context ...s of Grasp on the DIMACS benchmark problems [39]. Grasp is a greedy randomized adaptive search procedure that can find good quality solutions for a wide variety of combinatorial optimization problems =-=[7, 32, 8, 40]. In [39], four implementations of Grasp were applied to solve fi-=-ve classes of DIMACS SAT problems, "aim," "ii," "jnh," "ssa7552," and "par." Comparing to GSAT, Grasp did better on the "aim," "ssa7552... |

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Citation Context ...vercome the inefficiency of continuous unconstrained optimization methods, Gu developed discrete bit-parallel optimization algorithms (SAT 14.5 and SAT 14.6) to evaluate continuous objective function =-=[Gu94]-=- and have found significant performance improvements. 6 B. W. WAH AND Y. SHANG (b) Continuous Constrained Formulation. This generally involves a heuristic objective function that indicates the quality... |

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Citation Context ...ave also applied DLM to design multiplier-less QMF filter banks, which involves solving highly nonlinear discrete constrained optimization problems whose objectives and constraints are real functions =-=[55]-=-. To summarize, DLM is a generalization of local search schemes that optimize the objective alone and clause-weight schemes that optimize the constraints alone. When the search reaches a local minimum... |

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Citation Context ... a Lagrangian transformation does not reduce the number of local minima, and continuous Lagrangian methods are an order-of-magnitude more expensive to apply than the corresponding discrete algorithms =-=[CW95]-=-. 3. Discrete Lagrangian Methods for Solving SAT Problems As discussed in the last section, we formulate SAT problems as constrained optimization problems (1.2) and solve them using Lagrangian methods... |

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