. 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 NP-complete 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 ...

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 worst-case 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 worst-case upper bounds 2^{0.30897K} and 2^{0.10299L}, respectively.

The maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of approximation (polynomial-time) algorithms and exact (exponential-time) algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT 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 (MAX-SNP-complete) maximum cut problem (MAX-CUT), 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.

Recently there was a signicant progress in proving (exponential-time) worst-case upper bounds for the propositional satisability problem (SAT). MAX-SAT is an important generalization of SAT. Several upper bounds were obtained for MAX-SAT and its NP-complete subproblems. In particular, Niedermeier and Rossmanith recently proved the worst-case upper bound O 2 K=2:88::: for MAX-2-SAT (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 2-clauses. 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 2-clauses (and not 1-clauses). 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 ...

We survey recent algorithms for the propositional satisfiability problem, in particular algorithms that have the best current worst-case upper bounds on their complexity. We also discuss some related issues: the derandomization of the algorithm of Paturi, Pudlák, Saks and Zane, the Valiant-Vazirani Lemma, and random walk algorithms with the "back button".

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 NP-complete. It formalizes the spare allocation problem for reconfigurable arrays, an important problem from VLSI manufacturing.

Recently there was an explosion in proving (exponential-time) worst-case upper bounds for the propositional satisfiability problem (SAT) and related problems, mainly, k-SAT, MAX-SAT and MAX-2-SAT. The previous version of this paper contained an algorithm for MAX-2-SAT, 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 2-clauses 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 1-clauses). However, the use of Yannakakis' symmetric flow algorithm is replaced by several transformation rules.

Recent time has seen quite some progress in the development of exponential time algorithms for NP-hard 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 21--27, 1998. y Supported by a Feodor Lynen fellowship of the Alex...

We describe approximation algorithms for (unweighted) MAX SAT with performance ratios arbitrarily close to 1, in particular, when performance ratios exceed the limits of polynomial-time approximation. Namely, given a polynomial-time -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...

In this paper we present relatively simple proofs of the following new upper bounds. • c N, where c < 2 is a constant, N is the number of variables, for MAX-SAT for formulas with constant clause density; • 2 K/6, where K is the number of clauses, for MAX-2-SAT; • 2 N/6.7 for (n, 3)-MAX-2-SAT. All bounds are proved by the splitting method. The main two techniques are combined complexity measures and clause learning.