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A General Method to Speed Up FixedParameterTractable Algorithms
, 1999
"... A xedparametertractable algorithm, or FPT algorithm for short, gets an instance (I; k) as its input and has to decide whether (I; k) 2 L for some parameterized problem L. Many parameterized algorithms work in two stages: reduction to a problem kernel and bounded search tree. Their time complexity ..."
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Cited by 46 (18 self)
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A xedparametertractable algorithm, or FPT algorithm for short, gets an instance (I; k) as its input and has to decide whether (I; k) 2 L for some parameterized problem L. Many parameterized algorithms work in two stages: reduction to a problem kernel and bounded search tree. Their time complexity is then of the form O(p(jIj) + q(k) k ), where q(k) is the size of the problem kernel. We show how to modify these algorithms to obtain time complexity O(p(jIj) + k ), if q(k) is polynomial. Key words: Algorithms, Parametrized Complexity 1 Introduction A parameterized problem usually consists of two componentsthe input and aspects of the input that constitute a parameter. For example, the NPcomplete Vertex Cover problem has an undirected graph G as its input and a positive integer k as its parameter; the question is whether there is a set of at most k vertices that cover all edges in G. The central question of parameterized complexity theory [5] is as follows: Given a parameter...
Upper Bounds for Vertex Cover Further Improved
"... . The problem instance of Vertex Cover consists of an undirected graph G = (V; E) and a positive integer k, the question is whether there exists a subset C V of vertices such that each edge in E has at least one of its endpoints in C with jCj k. We improve two recent worst case upper bounds fo ..."
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Cited by 43 (16 self)
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. The problem instance of Vertex Cover consists of an undirected graph G = (V; E) and a positive integer k, the question is whether there exists a subset C V of vertices such that each edge in E has at least one of its endpoints in C with jCj k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn + 1:32472 k k 2 ), where n is the number of vertices in G. Afterwards, Downey et al. improved this to O(kn+ 1:31951 k k 2 ). Bringing the exponential base significantly below 1:3, we present the new upper bound O(kn + 1:29175 k k 2 ). 1 Introduction Vertex Cover is a problem of central importance in computer science: { It was among the rst NPcomplete problems [7]. { There have been numerous eorts to design ecient approximation algorithms [3], but it is also known to be hard to approximate [1]. { It is of central importance in parameterized complexity theory and has one ...
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.
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.
A New Algorithm for MAX2SAT
 In Proceedings of 17th International Symposium on Theoretical Aspects of Computer Science, STACS 2000
, 1999
"... Recently there was a signicant progress in proving (exponentialtime) worstcase upper bounds for the propositional satisability problem (SAT). MAXSAT is an important generalization of SAT. Several upper bounds were obtained for MAXSAT and its NPcomplete subproblems. In particular, Niedermeier ..."
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Cited by 16 (3 self)
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Recently there was a signicant progress in proving (exponentialtime) worstcase upper bounds for the propositional satisability problem (SAT). MAXSAT is an important generalization of SAT. Several upper bounds were obtained for MAXSAT and its NPcomplete subproblems. In particular, Niedermeier and Rossmanith recently proved the worstcase upper bound O 2 K=2:88::: for MAX2SAT (i.e. each clause contains at most two variables), where K is the number of clauses. In this paper we improve this bound to O 2 K2=4 , where K 2 is the number of 2clauses. In addition, our algorithm and the proof are much simpler than those of Niedermeier and Rossmanith. The key ideas are to use the symmetric ow algorithm of Yannakakis and to count only 2clauses (and not 1clauses). 1 Introduction. SAT (the problem of satisability of a propositional formula in conjunctive normal form (CNF )) can be easily solved in time of the order 2 N , where N is the number of variables in the ...
An Efficient Exact Algorithm for Constraint Bipartite Vertex Cover
"... The "Constraint Bipartite Vertex Cover" problem (CBVC for short) is: given a bipartite graph G with n vertices and two positive integers k 1 ; k 2 , is there a vertex cover taking at most k 1 vertices from one and at most k 2 vertices from the other vertex set of G? CBVC is NPcomplete. It formalize ..."
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Cited by 9 (3 self)
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The "Constraint Bipartite Vertex Cover" problem (CBVC for short) is: given a bipartite graph G with n vertices and two positive integers k 1 ; k 2 , is there a vertex cover taking at most k 1 vertices from one and at most k 2 vertices from the other vertex set of G? CBVC is NPcomplete. It formalizes the spare allocation problem for reconfigurable arrays, an important problem from VLSI manufacturing.
Algorithms for SAT and Upper Bounds on Their Complexity
, 2001
"... We survey recent algorithms for the propositional satisfiability problem, in particular algorithms that have the best current worstcase upper bounds on their complexity. We also discuss some related issues: the derandomization of the algorithm of Paturi, Pudlák, Saks and Zane, the ValiantVazirani ..."
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Cited by 8 (2 self)
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We survey recent algorithms for the propositional satisfiability problem, in particular algorithms that have the best current worstcase upper bounds on their complexity. We also discuss some related issues: the derandomization of the algorithm of Paturi, Pudlák, Saks and Zane, the ValiantVazirani Lemma, and random walk algorithms with the "back button".
Some Prospects for Efficient Fixed Parameter Algorithms
 In Proc. of the 25th Conference on Current Trends in Theory and Practice of Informatics (SOFSEM’98), Springer, LNCS 1521
, 1998
"... Recent time has seen quite some progress in the development of exponential time algorithms for NPhard problems, where the base of the exponential term is fairly small. These developments are also tightly related to the theory of fixed parameter tractability. In this incomplete survey, we explain so ..."
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Cited by 6 (0 self)
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Recent time has seen quite some progress in the development of exponential time algorithms for NPhard problems, where the base of the exponential term is fairly small. These developments are also tightly related to the theory of fixed parameter tractability. In this incomplete survey, we explain some basic techniques in the design of efficient fixed parameter algorithms, discuss deficiencies of parameterized complexity theory, and try to point out some future research challenges. The focus of this paper is on the design of efficient algorithms and not on a structural theory of parameterized complexity. Moreover, our emphasis will be laid on two exemplifying issues: Vertex Cover and MaxSat problems. A shorter version of this paper appears as an invited talk in the proceedings of the 25th Conference on Current Trends in Theory and Practice of Informatics (SOFSEM'98), Springer, LNCS , held in Jasna, Slovakia, November 2127, 1998. y Supported by a Feodor Lynen fellowship of the Alex...
MAXSAT approximation beyond the limits of polynomialtime approximation
, 1998
"... We describe approximation algorithms for (unweighted) MAX SAT with performance ratios arbitrarily close to 1, in particular, when performance ratios exceed the limits of polynomialtime approximation. Namely, given a polynomialtime approximation algorithm A 0 , we construct an ( + )approximati ..."
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Cited by 6 (0 self)
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We describe approximation algorithms for (unweighted) MAX SAT with performance ratios arbitrarily close to 1, in particular, when performance ratios exceed the limits of polynomialtime approximation. Namely, given a polynomialtime approximation algorithm A 0 , we construct an ( + )approximation algorithm A. The algorithm A runs in time of the order c k , where k is the number of clauses in the input formula and c is a constant depending on . Thus we estimate the cost of improving a performance ratio. Similar constructions for MAX 2SAT and MAX 3SAT are also described. Taking known algorithms as A 0 (for example, the Karlo{ Zwick algorithm for MAX 3SAT), we obtain particular upper bounds on the running time of A. 1 Introduction In the MAX SAT problem we are given a Boolean formula in conjunctive normal form and we seek a truth assignment that maximizes the number of satised clauses. An approximation algorithm for MAX SAT is an algorithm that nds an assignment sa...
A 2 K/4 time Algorithm for MAX2SAT: Corrected Version
, 1999
"... Recently there was an explosion in proving (exponentialtime) worstcase upper bounds for the propositional satisfiability problem (SAT) and related problems, mainly, kSAT, MAXSAT and MAX2SAT. The previous version of this paper contained an algorithm for MAX2SAT, and a \proof" of the theorem s ..."
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Cited by 5 (3 self)
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Recently there was an explosion in proving (exponentialtime) worstcase upper bounds for the propositional satisfiability problem (SAT) and related problems, mainly, kSAT, MAXSAT and MAX2SAT. The previous version of this paper contained an algorithm for MAX2SAT, and a \proof" of the theorem stating that this algorithm runs in time of the order 2 K 2 =4 , where K 2 is the number of 2clauses in the input formula. This bound and the corresponding bound 2 L=8 (where L is the length of the input formula) are still the best known. However, Jens Gramm pointed out to the author that the algorithm in the previous revision of this paper had an error. In this revision of the paper, we present a corrected version of the algorithm and the proof of the same upper bound. The proof is still based on the key idea to count only 2clauses (and not 1clauses). However, the use of Yannakakis' symmetric flow algorithm is replaced by several transformation rules.