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By Rolf Niedermeier
A fixed-parameter is an set of rules that offers an optimum strategy to a combinatorial challenge. This research-level textual content is an application-oriented creation to the becoming and hugely topical zone of the advance and research of effective fixed-parameter algorithms for difficult difficulties. The publication is split into 3 components: a huge creation that gives the final philosophy and motivation; through insurance of algorithmic tools constructed through the years in fixed-parameter algorithmics forming the middle of the booklet; and a dialogue of the fundamental from parameterized hardness conception with a spotlight on W [1]-hardness, which parallels NP-hardness, then pointing out a few kinfolk to polynomial-time approximation algorithms, and polishing off with an inventory of chosen case reports to teach the wide variety of applicability of the provided method. geared toward graduate and learn mathematicians, programmers, set of rules designers and laptop scientists, the booklet introduces the fundamental concepts and effects and gives a clean view in this hugely leading edge box of algorithmic learn.
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Extra info for Invitation to fixed-parameter algorithms
Example 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.