Algorithms – ESA 2004

The talk will survey the rich toolkit of FPT algorithm design methodologies that has developed over the last 20 years, including “older” techniques such as well-quasi-ordering, bounded treewidth, color-coding, reduction to a problem kernel, and bounded search trees — which have continued to deepen and advance — as well as some recently developed approaches such as win/win’s, greedy localization, iterative compression, and crown decompositions. Only in the last few years has it become clear that there are two distinct theoretical “games” being played in the effort to devise better and better FPT algorithms for various problems, that is, algorithms with a running time of O(f(k)n), where c is a fixed constant, k is the parameter, and n is the total input size. The first of these games is the obvious one of improving the parameter cost function f(k). For example, the first FPT algorithm for Vertex Cover had a running time of 2n. After a series of efforts, the best currently known algorithm (due to Chandran and Grandoni, in a paper to be presented at IWPEC 2004), has a parameter cost function of f(k) = (1.2745)kk4. The second theoretical game being played in FPT algorithm design has to do with efficient kernelization. The naturality and universality of this game comes from the observation that a parameterized problem is FPT if and only if there is a polynomial time preprocessing (also called data reduction or kernelization) algorithm that transforms the input (x, k) into (x′, k′), where k′ ≤ k and |x′| ≤ g(k) for some kernel-bounding function g(k), and where of course (x, k) is a yes-instance if and only if (x′, k′) is a yes-instance. In other words, modulo polynomial-time pre-processing, all of the difficulty, and even the size of the input that needs to be considered, is bounded by a function of the parameter. The natural theoretical game from this point of view about FPT is to improve the kernel-bounding function g(k). For example, after strenous efforts, a kernel-bounding function of g(k) = 335k has recently been achieved (by Alber et al.) for the Planar Dominating Set problem. Some of the new techniques to be surveyed pertain to the f(k) game, some address the kernelization game, and some are relevant to both of these goals in improving FPT algorithms. One of the striking things about the “positive” toolkit of FPT algorithm design is how unsettled the area seems to be, with S. Albers and T. Radzik (Eds.): ESA 2004, LNCS 3221, pp. 1–2, 2004. c © Springer-Verlag Berlin Heidelberg 2004