## 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

11605 |
Johnson: Computers and Intractability: A guide to theory of NP hardness, Feeman and
- Garey, S
- 1979
(Show Context)
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... |

1131 |
A computation procedure for quantification theory
- Davis, Putman
- 1960
(Show Context)
Citation Context ... , . . . , y1 s1 ], . . . , F [xl 1 , . . . , xl ri ][yl 1 , . . . , yl si the algorithm, and then working on each of the formulas recursively. This technique is similar to the Davis-Putnam procedure =-=[18]-=-. If at any time in the algorithm F becomes empty while the parameter k is still positive, then we stop and report that no truth assignment for F satisfying k clauses exists. Also, at the beginning of... |

926 | Parameterized Complexity
- Downey, Fellows
- 1999
(Show Context)
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... |

206 |
BuiIding Expert Systems
- Hayes-Roth, Waterman, et al.
- 1983
(Show Context)
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... |

154 | Vertex cover: Further observations and further improvements
- Chen, Kanj, et al.
(Show Context)
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... |

151 | Logic and Databases: A Deductive Approach
- Gallaire, Minker, et al.
- 1984
(Show Context)
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... |

134 | Which problems have strongly exponential complexity
- Impagliazzo, Paturi, et al.
- 2001
(Show Context)
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), ... |

111 |
Algorithms for the maximum satisfiability problem
- Hansen, Jaumard
- 1990
(Show Context)
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... |

107 | Fixed parameter algorithms for Dominating Set and related problems on planar graphs
- Alber, Bodlaender, et al.
(Show Context)
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... |

95 |
Algorithms for maximum independent sets
- Robson
- 1986
(Show Context)
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... |

80 | A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT problems
- Borchers, Furman
(Show Context)
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... |

69 | Parameterized complexity: A framework for systematically confronting computational intractability
- Downey, Fellows, et al.
- 1997
(Show Context)
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 |
A deterministic (2 − 2/(k + 1)) n algorithm for k-SAT based on local search
- Dantsin, Goerdt, et al.
- 2003
(Show Context)
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 |
Parameterizing above guaranteed values: MaxSat and MaxCut
- Mahajan, Raman
- 1999
(Show Context)
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... |

55 | Finding maximum independent sets in sparse and general graphs
- Beigel
- 1999
(Show Context)
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 |
Fixed-parameter tractability and approximation of NP-hard optimization problems
- Cai, Chen
- 1993
(Show Context)
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
- Eppstein
- 2001
(Show Context)
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], ... |

44 | A general method to speed up fixedparameter-tractable algorithms
- Niedermeier, Rossmanith
(Show Context)
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
- Asano, Williamson
- 2002
(Show Context)
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
- Bansal, Raman
- 1999
(Show Context)
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
- Alber, Fan, et al.
- 2005
(Show Context)
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
- Niedermeier, Rossmanith
(Show Context)
Citation Context |

36 | Deciding propositional tautologies: algorithms and their complexity
- Kullmann, Luckhardt
- 1997
(Show Context)
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
- Balasubramanian, Fellows, et al.
- 1998
(Show Context)
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
- Battiti, Protasi
- 1998
(Show Context)
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
- Gramm, Hirsch, et al.
- 2000
(Show Context)
Citation Context |

20 | Comparative studies of constraint satisfaction and Davis-Putnam algorithms for maximum satisfiability problems
- Wallace, Freuder
(Show Context)
Citation Context |

13 |
Integrity Constraints in logic databases
- Asirelli, Martelli
- 1989
(Show Context)
Citation Context |

13 |
Faster exact solutions for Max2Sat
- Gramm, Niedermeier
- 2000
(Show Context)
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
- Chen, Kanj
(Show Context)
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... |

9 |
A theoretical framework of hybrid approaches to max sat
- Asano, Hori, et al.
- 1997
(Show Context)
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
- Fernau, Niedermeier
(Show Context)
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... |

6 | Improvement on Vertex Cover for low-degree graphs, Networks 35
- Chen, Liu, et al.
- 2000
(Show Context)
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
- Wallace
- 1996
(Show Context)
Citation Context |

2 | Reactive research, a history base heuristic for Max-Sat - Battiti, Protasi - 1997 |

2 | Using nondeterminism to design efficient deterministic algorithms, Algorithmica 40
- Chen, Friesen, et al.
- 2004
(Show Context)
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 ... |