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

Abstract

We provide first-time fixed-parameter tractability results for the NPhard problems Maximum Full-Degree Spanning Tree and MinimumVertex Feedback Edge Set. These problems are dual to each other: In Maximum Full-Degree Spanning Tree, the task is to find a spanning tree for a given graph that maximizes the number of vertices that preserve their degree. For Minimum-Vertex Feedback Edge Set, the task is to minimize the number of vertices that end up with a reduced degree. Parameterized by the solution size, we exhibit that Minimum-Vertex Feedback Edge Set is fixed-parameter tractable and has a problem kernel with the number of vertices linearly depending on the parameter k. Our main contribution for Maximum Full-Degree Spanning Tree, which is W[1]-hard, is a linear-size problem kernel when restricted to planar graphs. Moreover, we present a dynamic programming algorithm for graphs of bounded treewidth.

(Meta) Kernelization

“…On the other hand, Theorem 1.3 applies to r-quasi-coverable problems and yields linear kernels. That way, it unifies and implies results presented in [4,5,15,16,19,41,46,47,53,59,62] as a corollary.…”

“…The work of [5] triggered an explosion of papers on kernelization, and in particular on kernelization of problems on planar graphs. Combining the ideas of [5] with problem specific data reduction rules, kernels of linear sizes were obtained for a variety of parameterized problems on planar graphs including Connected Vertex Cover, Minimum Edge Dominating Set, Maximum Triangle Packing, Efficient Edge Dominating Set, Induced Matching, Full-Degree Spanning Tree, Feedback Vertex Set, Cycle Packing, and Connected Dominating Set [3,5,15,16,19,46,47,53,59,62]. Dominating Set has received special attention from kernelization view point, leading to a linear kernel on graphs of bounded genus [41] and polynomial kernel on graphs excluding a fixed graph H as a minor and on d-degenerated graphs [6,64].…”

A Retrospective on (Meta) Kernelization

“…The first results of this kind appeared in [14] and they essentially initiated the research on meta-algorithmic for kernelization (see also [8] for an earlier version). The results in [14] subsumed several previous results on kernelization for problems on planar graphs such as [3,4,15,16,18,19,62,69,70,75,80,82]. Moreover, the algorithmic techniques in [14] introduced new concepts and tools that were of use in later investigations [42, 54, 55, 59-61, 74, 76, 77, 87].…”

A Linear Kernel for Planar Connected Dominating Set

“…These ideas were later abstracted by Guo and Niedermeier [15] who gave a framework in which to obtain linear kernels for planar graph problems possessing a certain ''locality property''. This framework has been successfully applied to yield linear kernels for the Connected Vertex Cover, Minimum Edge Dominating Set, Maximum Triangle Packing, Efficient Edge Dominating Set, Induced Matching and Full-Degree Spanning Tree problems [15,16,18]. However, the framework proposed by Guo and Niedermeier [15] in its current form is not able to handle problems like Feedback Vertex Set and Odd Cycle Transversal because these do not admit the ''locality property'' required by the framework.…”