Abstract
Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [(Meta) kernelization, in Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, 2009, pp. 629--638] on graphs of bounded genus, then generalized by Fomin et al. [Bidimensionality and kernels, in Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadephia, 2010, pp. 503--510] to graphs excluding a fixed minor, and by Kim et al. [Linear kernels and single-exponential algorithms via protrusion decompositions, in Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Comput. Sci., 7965 (2013), pp. 613--624] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions, but, mainly due to their generality, i...
Highlights
We essentially substitute the algorithmic power of Counting Monadic Second Order (CMSO) logic with that of dynamic programming on graphs of bounded decomposability
We propose a general definition of a problem encoding for the tables of dynamic programming when solving parameterized problems on graphs of bounded treewidth
While our framework can be seen as a possible formalization of dynamic programming, our purpose is to use it for constructing protrusion replacement algorithms and linear kernels whose size is explicitly determined
Summary
© Valentin Garnero, Christophe Paul, Ignasi Sau, and Dimitrios M. The main reason behind this non-constructibility is that the proofs rely on a property of problems called Finite Integer Index (FII) that, roughly speaking, allows to replace large “protrusions” (i.e., large subgraphs with small boundary to the rest of the graph) with “equivalent” subgraphs of constant size. This substitution procedure is known as protrusion replacer, and while its existence has been proved, so far, there is no generic way to construct it. This approach is essentially based on extensions of Courcelle’s theorem [4] that, even when they offer constructibility, it is hard to extract from them any explicit constant that upper-bounds the size of the derived kernel
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