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By Kenneth A. Berman, Jerome L. Paul
Reflecting the expanding value of parallel algorithms and parallel desktop architectures, this article presents in-depth assurance of conventional and present subject matters in sequential algorithms, in addition to an exceptional starting place within the concept of parallel algorithms.
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Sample text
From left to right, the three central parameterized problems Vertex Cover, Independent Set, and Dominating Set with optimal solution sets marked. Their parameterized complexity ranges from FPT over W [1 ]-complete to W [2 ]-complete. The parameter is always defined as the size of the solution set and the respective solution sets are marked by light shading (opposite to the dark shading of the remaining vertices). on the world of fixed-parameter tractable problems such as those exhibited by Vertex Cover.
1 that there is usually more than one natural way to parameterize the considered problem. As we will now see, it cannot be taken for granted that we will find the “right” parameterization for a problem. As a matter of fact, several reasonable and “equally valid” parameters to choose may exist, and which parameterization is to be preferred often depends on the application or on additional knowledge about the problem—if any is to be preferred at all. In what follows, we try to collect a sort of check list of questions to ask when confronted with a new problem where the goal is to solve it exactly by means of a fixed-parameter algorithm using a suitable parameterization.
1996). The idea of parameterizing above guaranteed values was put forward in Mahajan and Raman (1999), there restricted to the problems Maximum Satisfiability and Maximum Cut. How to approximate Weighted Vertex Cover with a ratio 2 is shown in Bar-Yehuda and Even (1981). This can be (asymptotically) improved to an approximation ratio 2−log log |V |/(2 log |V |) (Bar-Yehuda and Even, 1985; Monien and Speckenmeyer, 1985). Baker (1994) contains often cited PTAS results for Vertex Cover and several other problems restricted to planar graphs.