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56
Vertex Cover: Further Observations and Further Improvements
 Journal of Algorithms
, 1999
"... Recently, there have been increasing interests and progresses in lowering the worst case time complexity for wellknown NPhard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are int ..."
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Cited by 154 (15 self)
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Recently, there have been increasing interests and progresses in lowering the worst case time complexity for wellknown NPhard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are introduced, which lead to an improved algorithm of time O(kn + 1:271 k k 2 ) for the problem. Our algorithm also induces improvement on previous algorithms for the Independent Set problem on graphs of small degree. 1 Introduction Many optimization problems from industrial applications are NPhard. According to the NPcompleteness theory [10], these problems cannot be solved in polynomial time unless P = NP. However, this fact does not obviate the need for solving these problems for their practical importance. There has been a number of approaches to attacking the NPhardness of optimization problems, including approximation algorithms, heuristic algorithms, and average time analysis. Recent...
An Improved Exponentialtime Algorithm for kSAT
, 1998
"... We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, ..."
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Cited by 88 (5 self)
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We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment. We show that, for each k, the running time of ResolveSat on a kCNF formula is significantly better than 2 n , even in the worst case. In particular, we show that the algorithm finds a satisfying assignment of a general satisfiable 3CNF in time O(2 :448n ) with high probability; where the best previous algorithm [13] has running time O(2 :562n ). We obtain a better upper bound of 2 (2 ln 2\Gamma1)n+o(n) = O(2 0:387n ) for 3CNF that have exactly one satisfying assignment (unique kSAT). For each k, the bounds for general kCNF are the best currently known for ...
Improved Algorithms for 3Coloring, 3EdgeColoring, and Constraint Satisfaction
, 2001
"... We consider worst case time bounds for NPcomplete problems including 3SAT, 3coloring, 3edgecoloring, and 3list coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3SAT is equivalent to (2, 3)CSP while the other problems above are special cases ..."
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Cited by 47 (2 self)
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We consider worst case time bounds for NPcomplete problems including 3SAT, 3coloring, 3edgecoloring, and 3list coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3SAT is equivalent to (2, 3)CSP while the other problems above are special cases of (3, 2)CSP. We give a fast algorithm for (3, 2) CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of DavisPutnamstyle backtracking with more sophisticated matching and network flow based ideas. 1 Introduction There has recently been growing interest in analysis of superpolynomialtime algorithms, including algorithms for NPhard problems such as satisfiability or graph coloring. This interest has multiple causes: . Many important applications can be modeled with these problems, and with the increased speed of modern computers, solved effectively; for instance it is now routine to solve hard 500variable satisfia...
Improved upper bounds for 3sat
 In 15th ACMSIAM Symposium on Discrete Algorithms (SODA 2004). ACM and SIAM
"... The CNF Satisfiability problem is to determine, given a CNF formula F, whether or not there exists a satisfying assignment for F. If each clause of F contains at most k literals, then F is called a kCNF formula and the problem is called kSAT. For small k’s, especially for k = 3, there exists a lot ..."
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Cited by 42 (1 self)
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The CNF Satisfiability problem is to determine, given a CNF formula F, whether or not there exists a satisfying assignment for F. If each clause of F contains at most k literals, then F is called a kCNF formula and the problem is called kSAT. For small k’s, especially for k = 3, there exists a lot of algorithms which run significantly faster than the trivial 2n bound. The following list summarizes those algorithms where a constant c means that the algorithm runs in time O(cn). Roughly speaking most algorithms are based on DavisPutnam. [Sch99] is the first local search algorithm which gives a guaranteed performance for general instances and [DGH+02], [HSSW02], [BS03] and [Rol03] follow up this Schöning’s approach. 3SAT 4SAT 5SAT 6SAT type ref. 1.782 1.835 1.867 1.888 det. [PPZ97]
New WorstCase Upper Bounds for SAT
 Journal of Automated Reasoning
, 2000
"... In 1980 Monien and Speckenmeyer proved that satisfiability of a propositional formula consisting of K clauses (of arbitrary length) can be checked in time of the order 2^{K/3}. Recently Kullmann and Luckhardt proved the worstcase upper bound 2^{L/9}, where L is the length of the input formula. The ..."
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Cited by 35 (8 self)
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In 1980 Monien and Speckenmeyer proved that satisfiability of a propositional formula consisting of K clauses (of arbitrary length) can be checked in time of the order 2^{K/3}. Recently Kullmann and Luckhardt proved the worstcase upper bound 2^{L/9}, where L is the length of the input formula. The algorithms leading to these bounds are based on the splitting method which goes back to the Davis{Putnam procedure. Transformation rules (pure literal elimination, unit propagation etc.) constitute a substantial part of this method. In this paper we present a new transformation rule and two algorithms using this rule. We prove that these algorithms have the worstcase upper bounds 2^{0.30897K} and 2^{0.10299L}, respectively.
A probabilistic 3SAT algorithm further improved
, 2002
"... In [Sch99], Schöning proposed a simple yet ecient randomized algorithm for solving the k SAT problem. In the case of 3SAT, the algorithm has an expected running time of poly(n) (4=3) O(1:3334 ) when given a formula F on n variables. This was the up to now best running time known for an algo ..."
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Cited by 32 (2 self)
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In [Sch99], Schöning proposed a simple yet ecient randomized algorithm for solving the k SAT problem. In the case of 3SAT, the algorithm has an expected running time of poly(n) (4=3) O(1:3334 ) when given a formula F on n variables. This was the up to now best running time known for an algorithm solving 3SAT. In this paper, we describe an algorithm which improves upon this time bound by combining an improved version of the above randomized algorithm with other randomized algorithms. Our new expected time bound for 3SAT is O(1:3302 ).
Recovering and exploiting structural knowledge from cnf formulas
 In: Principles and Practice of Constraint Programming. Number 2470 in Lecture Notes in Computer Science
, 2002
"... Abstract. In this paper, a new preprocessing step is proposed in the resolution of SAT instances, that recovers and exploits structural knowledge that is hidden in the CNF. It delivers an hybrid formula made of clauses together with a set of equations of the form y = f(x1,...,xn) where f is a stand ..."
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Cited by 29 (1 self)
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Abstract. In this paper, a new preprocessing step is proposed in the resolution of SAT instances, that recovers and exploits structural knowledge that is hidden in the CNF. It delivers an hybrid formula made of clauses together with a set of equations of the form y = f(x1,...,xn) where f is a standard connective operator among (∨, ∧, ⇔) andwherey and xi are boolean variables of the initial SAT instance. This set of equations is then exploited to eliminate clauses and variables, while preserving satisfiability. These extraction and simplification techniques allowed us to implement a new SAT solver that proves to be the most efficient current one w.r.t. several important classes of instances.
Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems
 Algorithmica
, 2004
"... We present a framework for an automated generation of exact search tree algorithms for NPhard problems. The purpose of our approach is twofoldrapid development and improved upper bounds. ..."
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Cited by 24 (10 self)
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We present a framework for an automated generation of exact search tree algorithms for NPhard problems. The purpose of our approach is twofoldrapid development and improved upper bounds.
Deterministic algorithms for kSAT based on covering codes and local search
 Proceedings of the 27th International Colloquium on Automata, Languages and Programming, ICALP'2000, volume 1853 of Lecture Notes in Computer Science
, 2000
"... Abstract. We show that satisfiability of formulas in kCNF can be decided deterministically in time close to (2k/(k + 1)) n, where n is the number of variables in the input formula. This is the best known worstcase upper bound for deterministic kSAT algorithms. Our algorithm can be viewed as a dera ..."
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Cited by 23 (10 self)
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Abstract. We show that satisfiability of formulas in kCNF can be decided deterministically in time close to (2k/(k + 1)) n, where n is the number of variables in the input formula. This is the best known worstcase upper bound for deterministic kSAT algorithms. Our algorithm can be viewed as a derandomized version of Schöning’s probabilistic algorithm presented in [15]. The key point of our algorithm is the use of covering codes together with local search. Compared to other “weakly exponential ” algorithms, our algorithm is technically quite simple. We also show how to improve the bound above by moderate technical effort. For 3SAT the improved bound is 1.481 n. 1
New WorstCase Upper Bounds for MAX2SAT with Application to MAXCUT
, 2000
"... The maximum 2satisfiability problem (MAX2SAT) is: given a Boolean formula in 2CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX2SAT is MAXSNPcomplete. Recently, this problem received much attention in the contexts of approximation (polynomialtime) a ..."
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Cited by 22 (7 self)
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The maximum 2satisfiability problem (MAX2SAT) is: given a Boolean formula in 2CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX2SAT is MAXSNPcomplete. Recently, this problem received much attention in the contexts of approximation (polynomialtime) algorithms and exact (exponentialtime) algorithms. In this paper, we present an exact algorithm solving MAX2SAT in time poly(L) 2^(K/5), where K is the number of clauses and L is their total length. Since, in our analysis, we count only clauses containing exactly two literals, this bound implies the bound poly(L) 2^(L/10). Our results significantly improve previous bounds: poly(L) 2^(K/2.88) [30] and poly(L) 2^(K/3.44) (implicit in [4]). As an application, we derive upper bounds for the (MAXSNPcomplete) maximum cut problem (MAXCUT), showing that it can be solved in time poly(M) 2^(M/3), where M is the number of edges in the given graph. This is of special interest for graphs with low vertex degree.