Approximating Partially Bounded Degree Deletion on Directed Graphs

Abstract

The $\rm{B\small{OUNDED}}$ $\rm{D\small{EGREE}}$ $\rm{D\small{ELETION}}$ problem (BDD) is that of computing a minimum vertex set in a graph $G=(V, E)$ with degree bound $b: V\rightarrow\mathbb{Z}_+$, such that, when it is removed from $G$, the degree of any remaining vertex $v$ is no larger than $b(v)$. It is a classic problem in graph theory and various results have been obtained including an approximation ratio of $2+\ln b_{\max}$ [Okun, Barak, IPL, 2003], where $b_{\max}$ is the maximum degree bound. This paper considers BDD on directed graphs containing unbounded vertices, which we call $\rm{P\small{ARTIAL}}$ $\rm{B\small{OUNDED}}$ $\rm{D\small{EGREE}}$ $\rm{D\small{ELETION}}$ (PBDD). Despite such a natural generalization of standard BDD, it appears that PBDD has never been studied and no algorithmic results are known, approximation or parameterized. It will be shown that 1) in case all the possible degrees are bounded, in-degrees by $b^-$ and out-degrees by $b^+$, BDD on directed graphs can be approximated within $2+\max_{v\in V}\ln(b^-(v)+b^+(v))$, 2) it becomes NP-hard to approximate PBDD better than $b_{\max}$ (even on undirected graphs) once unbounded vertices are allowed, and 3) PBDD can be approximated within $\max\{2,b_{\max}+1\}$ when only in-degrees are partially bounded by $b$ (and the out-degrees of all the vertices are unbounded).