A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.

. 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 (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved in time O(|F | 1.3803 K ), where |F | is the length of a formula F in conjunctive normal form and K is the number of clauses in F . We also prove the time bounds O(|F |1.3995 k ), where k is the maximum number of satisfiable clauses, and O(1.1279 |F | ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K ). # An extended abstract of this paper was presented at the 26th International Colloquium on Automata, Languages, and Programming (ICALP'99), LNCS 1644, Springer-Verlag, pages 575--584, held in Prague, Czech Republic, July 11-15, 1999. + Supported by a Feodor Lynen fellowship (1998) of the Alexander von HumboldtStiftung, Bonn, and the Center for Discrete Ma...

We present a framework for an automated generation of exact search tree algorithms for NP-hard problems. The purpose of our approach is two-fold---rapid development and improved upper bounds.

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.

The maximum satisfiability problem (MAX-SAT) is stated as follows: Given Boolean formula in CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-SAT is MAX-SNP-complete and received much attention recently. One of the challenges posed by Alber, Gramm and Niedermeier in a recent survey paper asks: Can MAX-SAT be solved in less than 2 ' "steps"? Here, n is the number of different variables in the formula and a step may take polynomial time of the input. We answered this challenge positively by showing that popular algorithm based on branch-and-bound is bounded by O(b2 ') in time, where b is the maximum number of occurrences of any variable in the input.

We describe exact algorithms that provide new upper bounds for the Maximum Satisfiability problem (MaxSat). We prove that MaxSat can be solved in time O(|F | · 1.3972 K), where |F | is the length of a formula F in conjunctive normal form and K is the number of clauses in F. We also prove the time bounds O(|F | · 1.3995 k), where k is the maximum number of satisfiable clauses, and O((1.1279) |F | ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K). An exponential time approximation algorithm by Dantsin et al. uses an exact algorithm for MaxSat as a building block and is therefore also improved.

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

We consider the parameterized complexity of the following problem under the flamework introduced by Downey and Fellows[4]: Given a graph G, an integer parmneter : and a non-trivial hereditary property H, are there vertices of G that induce a subgraph with property H? This problem has been proved NP-hard by Lewis and Yanna- kakis[9]. e show that if H includes all independent sets but not all cliques or vice versa, then the problem is hard for the parameterized class kV[1] and is fixed parameter tractable otherwise. In the ibrmer case, if the tbrbidden set of the property is finite, we show, in fact, that the probleln is W[1]-complete (see [] for definitions). Our prooil, both of the tractability as well as the hardness ones, involve clever use of Ramsey nmnbers.