Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

Abstract

Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving N P-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = N P, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363-384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.Key words. parameterized algorithm, planar graph, dominating set, vertex cover, independent set, kernel AMS subject classifications. 05C85, 68Q17 DOI. 10.1137/050646354 1. Introduction. Many practical algorithms for N P-hard problems start by applying data reduction subroutines to the input instances of the problem. The hope is that after the data reduction phase the instance of the problem has shrunk to a moderate size. This makes the applicability of a second phase, such as a branchand-bound phase, to the resulting instance more feasible. Weihe showed in [41] how a practical preprocessing algorithm for a variation of the dominating set problem, called the red/blue dominating set problem, resulted in breaking up input instances of the problem into much smaller instances. Langston, and Shanbhag [2], in their implementation of algorithms for the vertex cover problem,

Fixed-parameter tractability results for full-degree spanning tree and its dual

“…Our main technical contribution is the proof of the linearsize problem kernel for FDST in planar graphs. It is easily conceivable that there is room for significantly improving the involved worst-case constant factors by refined mathematical analysis-the situation is comparable (but perhaps even more technical) with analogous results for Dominating Set in planar graphs [2,9]. Having obtained linear problem kernel sizes for a problem and its dual, the way for applying the lower bound technique for kernel size due to Chen et al [9] now seems open.…”

“…We remark that, when restricted to planar graphs, this work amends the so far few examples where both a problem and its dual possess linear-size problem kernels. The only other examples we are aware of (again restricted to planar graphs) are Vertex Cover and its dual Independent Set, and Dominating Set and its dual Nonblocker [2,[9][10][11].…”

“…More precisely, it must even be outerplanar, that is, it has a crossing-free embedding in the plane such that all vertices are on the same face, but this detail does not need to concern us here 9. Analogously to above, this graph is actually outerplanar.…”

Rank Vertex Cover as a Natural Problem for Algebraic Compression

“…On the other hand, in the context of kernelization, the picture is clear in the following sense: It is known that Vertex Cover admits a kernel with O(k 2 ) vertices and edges [2], but unless NP ⊆ co-NP/poly, it does not admit a kernel with O(k 2−ǫ ) edges [9]. We remark that it is also known that Vertex Cover admits a kernel not only of size O(k 2 ), but also with only 2k vertices [5,20], and it is conjectured that this bound might be essentially tight [4].…”

Diminishable parameterized problems and strict polynomial kernelization

“…2 They also showed a similar result for CNF-Sat parameterized by the number of variables. Both of these results seemingly are the only known polynomial kernel lower bounds that rely on the assumption of P = NP (see Chen et al [9] for a few linear lower bounds that also rely on P = NP). The goal of this paper is to show that Chen et al's framework applies for more problems, indeed allowing for a surprisingly simple, natural, and elegant proof framework.…”