Abstract.To determine if a graph has a spanning tree with few leaves is NP-hard as HAMIL-TONIAN PATH is a special case. In this paper we study the parametric dual of this problem, k-INTERNAL SPANNING TREE (Does G have a spanning tree with at least k internal vertices?). We give an algorithm running in time. We also give a 2-approximation algorithm for the problem.However, the main contribution of this paper is that we show the following remarkable structural bindings between k-INTERNAL SPANNING TREE and k-VERTEX COVER:• NO for k-VERTEX COVER implies YES for k-INTERNAL SPANNING TREE.• YES for k-VERTEX COVER implies NO for (2k + 1)-INTERNAL SPANNING TREE.We give a polynomial-time algorithm that produces either a vertex cover of size k or a spanning tree with at least k internal vertices. We show how to use this inherent vertex cover structure to design algorithms for FPT problems, here illustrated mainly by k-INTERNAL SPANNING TREE. We also briefly discuss the application of this vertex cover methodology to the parametric dual of the DOMINATING SET problem. This design technique seems to apply to many other FPT problems.

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Abstract. We deal with two important problems in pattern recognition that arise in the analysis of large datasets. While most feature subset selection methods use statistical techniques to preprocess the labeled datasets, these methods are generally not linked with the combinatorial properties of the final solutions. We prove that it is N P −hard to obtain an appropriate set of thresholds that will transform a given dataset into a binary instance of a robust feature subset selection problem. We address this problem using an evolutionary algorithm that learns the appropriate value of the thresholds. The empirical evaluation shows that robust subset of genes can be obtained. This evaluation is done using real data corresponding to the gene expression of lymphomas.

We consider the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H. For H = K 3 (the triangle) we give an O(2 2k log k+1.869k n 2 ) algorithm, and for general H an O(2 k|H| log k+2k|H| log |H| n |H| ) algorithm. We introduce a preprocessing (kernelization) technique based on crown decompositions of an auxiliary graph. For H = K 3 this leads to a preprocessing algorithm that reduces an arbitrary input graph of the problem to a graph on O(k 3 ) vertices in polynomial time. INTRODUCTIONFor a fixed graph H and an input graph G, the H-packing problem asks for the maximum number of vertex-disjoint copies of H in G. The K 2 -packing (edge packing) problem, which is equivalent to maximum matching, played a central role in the history of classical computational complexity. The first step towards the dichotomy of 'good' (polynomialtime) versus 'presumably-not-good' (NP-hard) was made in a paper on maximum matching from 1965 [E65], which gave a polynomial time algorithm for that problem. On the other hand, the K 3 -packing (triangle packing) problem, which is our main concern in this paper, is NP-hard [HK78].Recently, there has been a growing interest in the area of exact exponential-time algorithms for NP-hard problems. When measuring time in the classical way, simply by the 1

A survey of the most important and general techniques in parameterized algorithm design is given. Each technique is explained with a meta-algorithm, its use is illustrated by examples, and it is placed in a taxonomy under the four main headings of branching, kernelization, induction and win/win. This is the case with catalytic vertices # local reduction rules, the algorithmic method # extremal method, modelled crown reductions # crown reductions, FPT by treewidth # imposing FPT structure, reduction to independent set structure # imposing FPT structure, method of testsets # imposing FPT structure. 4 This is the case with interleaving, which combines bounded search trees and local reduction rules. The technique known as bounded integer linear THE