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**1 - 6**of**6**### Evolving Fixed-parameter Tractable Algorithms

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### Evolving Fixed-parameter Tractable Algorithms Stefan A. van der Meer a Iris van Rooij a,b Ida Sprinkhuizen-Kuyper a,b

"... One effective means of computing NP-hard problems is provided by fixed-parameter tractable (fpt-) algorithms. An fpt-algorithm is an algorithm whose running time is polynomial in the input size and superpolynomial only as a function of an input parameter. Provided that the parameter is small enough, ..."

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One effective means of computing NP-hard problems is provided by fixed-parameter tractable (fpt-) algorithms. An fpt-algorithm is an algorithm whose running time is polynomial in the input size and superpolynomial only as a function of an input parameter. Provided that the parameter is small enough, an fpt-algorithm runs fast even for large inputs. In this paper, we report on an investigation of the evolvability of fpt-algorithms via Genetic Programming (GP). The problem used in this investigation is the NP-hard 2D-Euclidean Traveling Salesman Problem (TSP), which is known to be fpt if the number of points not on the convex hull is taken as the parameter. The algorithm evolved in our GP study turns out to have clear characteristics of an fpt-algorithm. The results suggest GP can be utilized for generating fpt-algorithms for NP-hard problems in general, as well as for discovering input parameters that could be used to develop fpt-algorithms. 1

### unknown title

"... Compilability is a measure of how effectively compilation (or preprocessing) can be applied to knowledge bases specified in a particular knowledge representation formalism; the aim of compilation is to allow for efficient, on-line query processing. A theory of compilability has been established for ..."

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Compilability is a measure of how effectively compilation (or preprocessing) can be applied to knowledge bases specified in a particular knowledge representation formalism; the aim of compilation is to allow for efficient, on-line query processing. A theory of compilability has been established for organizing knowledge representation formalisms according to a scheme of “compilability classes”, and bears strong analogies to the classical theory of complexity, which permits the organization of computational problems according to complexity classes. We develop a novel theory of compilability, called parameterized compilability, which incorporates the notion of parameterization as used in parameterized complexity and permits for refined analysis of compilability. 1

### Efficient Implementation of the BSP/CGM Parallel Vertex Cover FPT Algorithm ⋆

"... Abstract. In many applications NP-complete problems need to be solved exactly. One promising method to treat some intractable problems is by considering the so-called Parameterized Complexity that divides the problem input into a main part and a parameter. The main part of the input contributes poly ..."

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Abstract. In many applications NP-complete problems need to be solved exactly. One promising method to treat some intractable problems is by considering the so-called Parameterized Complexity that divides the problem input into a main part and a parameter. The main part of the input contributes polynomially on the total complexity of the problem, while the parameter is responsible for the combinatorial explosion. We consider the parallel FPT algorithm of Cheetham et al. to solve the k-Vertex Cover problem, using the CGM model. Our contribution is to present a refined and improved implementation. In our parallel experiments, we obtained better results and obtained smaller cover sizes for some input data. The key idea for these results was the choice of good data structures and use of the backtracking technique. We used 5 graphs that represent conflict graphs of amino acids, the same graphs used also by Cheetham et al. in their experiments. For two of these graphs, the times we obtained were approximately 115 times better, for one of them 16 times better, and, for the remaining graphs, the obtained times were slightly better. We must also emphasize that we used a computational environment that is inferior than that used in the experiments of Cheetham et al.. Furthermore, for three graphs, we obtained smaller sizes for the cover. 1

### An Alternative Implementation for the FPT k-Vertex Cover Parallel Algorithm

, 2009

"... Adequate choice of data structures and special effort in implementation are crucial to the good performance of parallel algorithms. In this paper, we present experimental results of a BSP/CGM implementation for the FPT (Fixed-Parameter Tractability) Vertex Cover problem, also known as k-Vertex Cover ..."

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Adequate choice of data structures and special effort in implementation are crucial to the good performance of parallel algorithms. In this paper, we present experimental results of a BSP/CGM implementation for the FPT (Fixed-Parameter Tractability) Vertex Cover problem, also known as k-Vertex Cover. We propose an alternative implementation that has as its basis an algorithm that combines the parallel FPT algorithm proposed by Cheetham et al. and the Downey’s et al. sequential FPT algorithm. Previously, a better and refined implementation, based on the Cheetham et al. Algorithm was presented by Hanashiro. In his experiments, Hanashiro obtained better results than those presented by Cheetham et al. In this paper, implemented the new adapted algorithm for the k-Vertex Cover and compared our experimental results with those of Hanashiro et al, using the same input data (conflict graphs of amino acids). We report substantial improvement over the results of Hanahiro et al, with speedups from 3 to 20 times relative to that implementation.