Abstract
We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve an open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number.
Highlights
This paper studies the Bounded-Degree Vertex Deletion problem (BDD): given an undirected graph G, a degree bound d, and a limit, determine whether it is possible to delete at most vertices from G in order to obtain a graph of maximum degree at most d
On the way towards this result, we provide hardness results for several interesting versions of the multidimensional subset sum problem which we believe are interesting in their own right
Our results close a wide gap in the understanding of the complexity landscape of BDD parameterized by structural parameters
Summary
This paper studies the Bounded-Degree Vertex Deletion problem (BDD): given an undirected graph G, a degree bound d, and a limit , determine whether it is possible to delete at most vertices from G in order to obtain a graph of maximum degree at most d. Since the problem is NP-complete in general, it is natural to ask under which conditions does the problem become tractable In this direction, the parameterized complexity paradigm [12, 15, 41] allows a more refined analysis of the problem’s complexity than classical complexity. We associate each instance with a numerical parameter k and are most often interested in the existence of a fixed-parameter algorithm, i.e., an algorithm solving the problem in time f (k) ⋅ |V(G)|O(1) for some computable function f. Parameterized problems admitting such an algorithm belong to the class FPT; on the other hand, parameterized problems that are hard for the complexity class W[1] or W[2] do not admit fixed-parameter algorithms (under standard complexity assumptions)
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