Recently, there have been increasing interests and progresses in lowering the worst case time complexity for well-known NP-hard 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 NP-hard. 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 NP-hardness of optimization problems, including approximation algorithms, heuristic algorithms, and average time analysis. Recent...

We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the k-face cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.

We explore a method for computing admissible heuristic evaluation functions for search problems. It utilizes pattern databases (Culberson & Schaeffer, 1998), which are precomputed tables of the exact cost of solving various subproblems of an existing problem. Unlike standard pattern database heuristics, however, we partition our problems into disjoint subproblems, so that the costs of solving the different subproblems can be added together without overestimating the cost of solving the original problem. Previously (Korf & Felner, 2002) we showed how to statically partition the sliding-tile puzzles into disjoint groups of tiles to compute an admissible heuristic, using the same partition for each state and problem instance. Here we extend the method and show that it applies to other domains as well. We also present another method for additive heuristics which we call dynamically partitioned pattern databases. Here we partition the problem into disjoint subproblems for each state of the search dynamically. We discuss the pros and cons of each of these methods and apply both methods to three different problem domains: the sliding-tile puzzles, the 4-peg Towers of Hanoi problem, and finding an optimal vertex cover of a graph. We find that in some problem domains, static partitioning is most effective, while in others dynamic partitioning is a better choice. In each of these problem domains, either statically partitioned or dynamically partitioned pattern database heuristics are the best known heuristics for the problem.

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.

A xed-parameter-tractable 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 NP-complete 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...

Liming Cai School of EE & CS Ohio University Athens, OH 45701 USA Email:cai@leon.cs.ohiou.edu Fax:+1 740 593 0007 David Juedes School of EE & CS Ohio University Athens, OH 45701 USA Email:juedes@ohiou.edu Fax:+1 740 593 0007 Abstract It is shown that for essentially all MAX SNP-hard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max c-Sat, and a number of problems on bounded degree graphs such as Dominating Set and Independent Set, unless W [1] = FPT . Our results are derived via an approach that uses an extended parameterization of optimization problems and associated techniques to relate the parameterized complexity of problems in FPT to the parameterized complexity of extended versions that are W [1]-hard. Track: A Keywords: computational complexity, parameterized complexity, combinatorial optimization. # This work was supported by the National Science Foundation research grant CCR-000246 1

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We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition can be found in O ( � γ(G)n) time. The same technique can be used to show that the k-face cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved √ k in O(c1 n + n2) time, where c1 = 236√34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set. Keywords. NP-complete problems, fixed parameter tractability, planar graphs, planar dominating set, face cover, outerplanarity, treewidth.

Graph separation is a well-known tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many well-known NP-hard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p

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...