## Improved exact algorithms for MAX-SAT (2002)

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Venue: | Discrete Applied Mathematics |

Citations: | 16 - 1 self |

### BibTeX

@INPROCEEDINGS{Chen02improvedexact,

author = {Jianer Chen and Iyad A. Kanj},

title = {Improved exact algorithms for MAX-SAT},

booktitle = {Discrete Applied Mathematics},

year = {2002},

pages = {341--355},

publisher = {Springer}

}

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### Abstract

In this paper we present improved exact and parameterized algorithms for the maximum satisfiability problem. In particular, we give an algorithm that computes a truth assignment for a boolean formula F satisfying the maximum number of clauses in time O(1.3247 m |F |), where m is the number of clauses in F, and |F | is the sum of the number of literals appearing in each clause in F. Moreover, given a parameter k, we give an O(1.3695 k + |F |) parameterized algorithm that decides whether a truth assignment for F satisfying at least k clauses exists. Both algorithms improve the previous best algorithms by Bansal and Raman for the problem. Key words. maximum satisfiability, exact algorithms, parameterized algorithms. 1

### Citations

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Citation Context ...ked to decide if F has a truth assignment that satisfies at least k clauses. It is easy to see that both Max-Sat and parameterized Max-Sat are NP-hard since they generalize the satisfiability problem =-=[24]-=-. According to the theory of NP-completeness [24], these problems cannot be solved in polynomial time unless P = NP. The above fact does not diminish the need for solving these problems for their prac... |

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Citation Context ...m [21] (n is the number of vertices in the graph). A closely related line of research to the above one that has been receiving a lot of attention recently, is the area of fixed-parameter tractability =-=[19]-=-, which has found many applications in fields like databases, artificial intelligence, logic, and computational biology [20]. A famous problem that belongs to this category is the Vertex Cover problem... |

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Citation Context ...mportance. Due to its close relationship to the satisfiability problem (Sat), Max-Sat has many applications in artificial intelligence, combinatorial optimization, expert-systems, and databasesystems =-=[5, 8, 23, 27, 28, 32, 36, 37]-=-. Many approaches have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms [3, 4, 9], heuristic algorithms [8, 9, 10, 36], and exact and parameteriz... |

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Citation Context ...longs to this category is the Vertex Cover problem. Recently, there has been extensive research done aiming at improving the complexity of parameterized algorithms that solve the Vertex Cover problem =-=[6, 15, 16]-=-. Other examples of parameterized problems include Planar Dominating Set [1, 2], k-Dimensional Matching [13, 19], and Constraint Bipartite Vertex Cover [14, 22]. The Max-Sat problem has played a signi... |

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Citation Context ...mportance. Due to its close relationship to the satisfiability problem (Sat), Max-Sat has many applications in artificial intelligence, combinatorial optimization, expert-systems, and databasesystems =-=[5, 8, 23, 27, 28, 32, 36, 37]-=-. Many approaches have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms [3, 4, 9], heuristic algorithms [8, 9, 10, 36], and exact and parameteriz... |

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Citation Context ...er, have reduced significantly the number of cases in the problems considered. The general open question that is posed is to what extent we can keep improving these upper bounds? Using the results in =-=[29]-=-, one can easily show that if the Max-Sat problem can be solved in sub-linear exponential time (i.e., O(2 o(m) p(n)), where m is the number of clauses, n the number of variables, and p a polynomial), ... |

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Citation Context ...mportance. Due to its close relationship to the satisfiability problem (Sat), Max-Sat has many applications in artificial intelligence, combinatorial optimization, expert-systems, and databasesystems =-=[5, 8, 23, 27, 28, 32, 36, 37]-=-. Many approaches have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms [3, 4, 9], heuristic algorithms [8, 9, 10, 36], and exact and parameteriz... |

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Citation Context ...ve research done aiming at improving the complexity of parameterized algorithms that solve the Vertex Cover problem [6, 15, 16]. Other examples of parameterized problems include Planar Dominating Set =-=[1, 2]-=-, k-Dimensional Matching [13, 19], and Constraint Bipartite Vertex Cover [14, 22]. The Max-Sat problem has played a significant role in both these lines of research. Considerable efforts have been pai... |

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Citation Context ...algorithms. Numerous NP-hard combinatorial problems were studied from this point of view. Examples of exponential time algorithms for NP-hard optimization problems include the Independent Set problem =-=[11, 35]-=-, Dantsin et al.’s O(1.481 n ) time deterministic algorithm for the 3-Sat problem [17] (n is the number of variables in the formula), ∗ A preliminary version of this paper was presented at the 5th Lat... |

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Citation Context ...asesystems [5, 8, 23, 27, 28, 32, 36, 37]. Many approaches have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms [3, 4, 9], heuristic algorithms =-=[8, 9, 10, 36]-=-, and exact and parameterized algorithms [7, 12, 25, 31, 33]. In this paper we focus our attention on finding exact solutions for the Max-Sat problem. Acknowledging the apparent inevitability of an ex... |

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Citation Context ...ng a lot of attention recently, is the area of fixed-parameter tractability [19], which has found many applications in fields like databases, artificial intelligence, logic, and computational biology =-=[20]-=-. A famous problem that belongs to this category is the Vertex Cover problem. Recently, there has been extensive research done aiming at improving the complexity of parameterized algorithms that solve... |

66 |
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Citation Context ...Examples of exponential time algorithms for NP-hard optimization problems include the Independent Set problem [11, 35], Dantsin et al.’s O(1.481 n ) time deterministic algorithm for the 3-Sat problem =-=[17]-=- (n is the number of variables in the formula), ∗ A preliminary version of this paper was presented at the 5th Latin American Symposium on Theoretical Informatics (LATIN), and appeared in Lecture Note... |

64 |
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Citation Context ...hes have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms [3, 4, 9], heuristic algorithms [8, 9, 10, 36], and exact and parameterized algorithms =-=[7, 12, 25, 31, 33]-=-. In this paper we focus our attention on finding exact solutions for the Max-Sat problem. Acknowledging the apparent inevitability of an exponential time complexity for NP-hard problems, this line of... |

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Citation Context ...algorithms. Numerous NP-hard combinatorial problems were studied from this point of view. Examples of exponential time algorithms for NP-hard optimization problems include the Independent Set problem =-=[11, 35]-=-, Dantsin et al.’s O(1.481 n ) time deterministic algorithm for the 3-Sat problem [17] (n is the number of variables in the formula), ∗ A preliminary version of this paper was presented at the 5th Lat... |

45 |
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Citation Context ...hes have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms [3, 4, 9], heuristic algorithms [8, 9, 10, 36], and exact and parameterized algorithms =-=[7, 12, 25, 31, 33]-=-. In this paper we focus our attention on finding exact solutions for the Max-Sat problem. Acknowledging the apparent inevitability of an exponential time complexity for NP-hard problems, this line of... |

45 | Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction
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Citation Context ...venue, Chicago, IL 60604, ikanj@cs.depaul.edu. This work was supported in part by DePaul University Competitive Research Grant. 1sand Eppstein’s O(1.3289 n ) time algorithm for the 3-Coloring problem =-=[21]-=- (n is the number of vertices in the graph). A closely related line of research to the above one that has been receiving a lot of attention recently, is the area of fixed-parameter tractability [19], ... |

45 | A General Method to Speed Up FixedParameter-Tractable Algorithms
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Citation Context ...needed to reduce |F | to a boolean formula whose length is O(k2 ), and along each root-leaf path in the searching tree we spend O(k2 ) time. Using a technique introduced by Niedermeier and Rossmanith =-=[34]-=-, we can get rid of the size of the kernel in the running time of the algorithm. The running time of the algorithm becomes O(αk max + |F |). The following general branching rules prove to be useful in... |

40 | Improved Approximation Algorithms for MAX SAT
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Citation Context ...ation, expert-systems, and databasesystems [5, 8, 23, 27, 28, 32, 36, 37]. Many approaches have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms =-=[3, 4, 9]-=-, heuristic algorithms [8, 9, 10, 36], and exact and parameterized algorithms [7, 12, 25, 31, 33]. In this paper we focus our attention on finding exact solutions for the Max-Sat problem. Acknowledgin... |

40 |
Upper bounds for maxsat: Further improved
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Citation Context ...hes have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms [3, 4, 9], heuristic algorithms [8, 9, 10, 36], and exact and parameterized algorithms =-=[7, 12, 25, 31, 33]-=-. In this paper we focus our attention on finding exact solutions for the Max-Sat problem. Acknowledging the apparent inevitability of an exponential time complexity for NP-hard problems, this line of... |

37 | A refined search tree technique for Dominating Set on planar graphs
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Citation Context ...ve research done aiming at improving the complexity of parameterized algorithms that solve the Vertex Cover problem [6, 15, 16]. Other examples of parameterized problems include Planar Dominating Set =-=[1, 2]-=-, k-Dimensional Matching [13, 19], and Constraint Bipartite Vertex Cover [14, 22]. The Max-Sat problem has played a significant role in both these lines of research. Considerable efforts have been pai... |

37 | New upper bounds for maximum satisfiability
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Citation Context |

36 | Deciding propositional tautologies: algorithms and their complexity
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Citation Context ...e following recurrence relation for the size of the search tree C(k ′ ) rooted at x holds: C(k ′ r� ) = C(k i=1 ′ − k ′ i) It is well known that the solution to the above recurrence (see for instance =-=[30]-=-) is C(k ′ ) = O(αk′ ), where α is the unique positive root of the characteristic polynomial p(x) = xk′ r − xk′ r−k ′ 1 − . . . − x k′ r−k ′ r−1 − 1. Now the size of the whole search tree will be O(αk... |

36 | Checking an expert system’s knowledge base for consistency and completeness - Nguyen, Perkins, et al. - 1985 |

21 |
An improved fixed parameter algorithm for vertex cover
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Citation Context ...longs to this category is the Vertex Cover problem. Recently, there has been extensive research done aiming at improving the complexity of parameterized algorithms that solve the Vertex Cover problem =-=[6, 15, 16]-=-. Other examples of parameterized problems include Planar Dominating Set [1, 2], k-Dimensional Matching [13, 19], and Constraint Bipartite Vertex Cover [14, 22]. The Max-Sat problem has played a signi... |

21 | Approximate Algorithms and Heuristics for MaxSat
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Citation Context ...ation, expert-systems, and databasesystems [5, 8, 23, 27, 28, 32, 36, 37]. Many approaches have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms =-=[3, 4, 9]-=-, heuristic algorithms [8, 9, 10, 36], and exact and parameterized algorithms [7, 12, 25, 31, 33]. In this paper we focus our attention on finding exact solutions for the Max-Sat problem. Acknowledgin... |

21 | New worst-case upper bounds for MAX-2-SAT with application to MAX-CUT
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Citation Context ...and Rossmanith’s result and presented algorithms of running time O(1.3802 k k 2 + |F |) for parameterized Max-Sat, and O(1.341294 m |F |) and O(1.105729 |F | |F |) for Max-Sat. Recently, Gramm et al. =-=[25, 26]-=- considered the Max-2-Sat problem, a special case of the Max-Sat problem. They gave an O(1.148699 m |F |) algorithm for the problem. Their algorithm also implies better algorithms for the parameterize... |

12 | Constrained minimum vertex cover in bipartite graphs: complexity and parameterized algorithmics
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Citation Context ... that solve the Vertex Cover problem [6, 15, 16]. Other examples of parameterized problems include Planar Dominating Set [1, 2], k-Dimensional Matching [13, 19], and Constraint Bipartite Vertex Cover =-=[14, 22]-=-. The Max-Sat problem has played a significant role in both these lines of research. Considerable efforts have been paid trying to lower the worst-case complexity for the problem. Cai and Chen [12] pr... |

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Citation Context ...ation, expert-systems, and databasesystems [5, 8, 23, 27, 28, 32, 36, 37]. Many approaches have been employed in dealing with the NP-hardness of the Max-Sat problem including approximation algorithms =-=[3, 4, 9]-=-, heuristic algorithms [8, 9, 10, 36], and exact and parameterized algorithms [7, 12, 25, 31, 33]. In this paper we focus our attention on finding exact solutions for the Max-Sat problem. Acknowledgin... |

9 | An Efficient Exact Algorithm for Constraint Bipartite Vertex Cover
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Citation Context ... that solve the Vertex Cover problem [6, 15, 16]. Other examples of parameterized problems include Planar Dominating Set [1, 2], k-Dimensional Matching [13, 19], and Constraint Bipartite Vertex Cover =-=[14, 22]-=-. The Max-Sat problem has played a significant role in both these lines of research. Considerable efforts have been paid trying to lower the worst-case complexity for the problem. Cai and Chen [12] pr... |

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Citation Context ...longs to this category is the Vertex Cover problem. Recently, there has been extensive research done aiming at improving the complexity of parameterized algorithms that solve the Vertex Cover problem =-=[6, 15, 16]-=-. Other examples of parameterized problems include Planar Dominating Set [1, 2], k-Dimensional Matching [13, 19], and Constraint Bipartite Vertex Cover [14, 22]. The Max-Sat problem has played a signi... |

3 | Enhancing maximum satisfiability algorithms with pure literal strategies
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2 | Reactive research, a history base heuristic for Max-Sat - Battiti, Protasi - 1997 |

2 | Using nondeterminism to design efficient deterministic algorithms, Algorithmica 40
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Citation Context ...oving the complexity of parameterized algorithms that solve the Vertex Cover problem [6, 15, 16]. Other examples of parameterized problems include Planar Dominating Set [1, 2], k-Dimensional Matching =-=[13, 19]-=-, and Constraint Bipartite Vertex Cover [14, 22]. The Max-Sat problem has played a significant role in both these lines of research. Considerable efforts have been paid trying to lower the worst-case ... |