By Rolf Niedermeier
A fixed-parameter is an set of rules that offers an optimum option to a combinatorial challenge. This research-level textual content is an application-oriented creation to the turning out to be and hugely topical zone of the improvement and research of effective fixed-parameter algorithms for difficult difficulties. The booklet is split into 3 components: a vast creation that gives the final philosophy and motivation; via insurance of algorithmic tools built through the years in fixed-parameter algorithmics forming the middle of the ebook; and a dialogue of the basic from parameterized hardness thought with a spotlight on W -hardness, which parallels NP-hardness, then pointing out a few family members to polynomial-time approximation algorithms, and polishing off with a listing of chosen case stories to teach the big variety of applicability of the awarded technique. aimed toward graduate and learn mathematicians, programmers, set of rules designers and laptop scientists, the ebook introduces the elemental strategies and effects and offers a clean view in this hugely cutting edge box of algorithmic study.
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Additional resources for Invitation to fixed-parameter algorithms
We close with two remarks. First, note that all the above problems in which vertex cover structure was applied are derived from maximization problems with 38 VERTEX COVER—AN ILLUSTRATIVE EXAMPLE respect to parameter k. It would be interesting to see applications for minimization problems, where small parameter values seem more common. Second, the above scenario is not limited to vertex cover structure; something like “dominating set structure” etc. might be interesting as well. Such investigations are only at the very beginning and should be extended in future research.
Very recently, a still more general version, so-called Capacitated Vertex Cover, with applications in drug design, has been considered. Here, the point is that each vertex can only cover a pre-speciﬁed number of edges incident on it; this is called the capacity of a vertex. Hence choosing a vertex to be in the cover set may not suﬃce to cover all its incident edges. This problem has recently been shown to be ﬁxed-parameter tractable with respect to the parameter solution size; the corresponding combinatorial explosion (so far) is much worse than for Vertex Cover.
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. √ k Theoretical work on c -algorithms for Vertex Cover on planar graphs appears in Alber et al. (2003) and Alber et al. (2004)—accompanying empirical studies appear in Alber et al.