Approximating partially bounded degree deletion on directed graphs

The Bounded Degree Deletion problem (BDD) is that of computing a minimum vertex set in a graph G=(V,E)  with degree bound b: V → 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+lnb_max [30], where b_max is the maximum degree bound. This paper considers BDD on directed graphs containing unbounded vertices, which we call Partially Bounded Degree Deletion (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 ϵ}Vln(b−(v)+b⁺(v)), and (2) although it becomes NP-hard to approximate PBDD better than bmax (even on undirected graphs) once unbounded vertices are allowed, it can be within max-2,b_max+1} when only in-degrees (and none of out-degrees) are partially bounded by b (and the out-degrees of all the vertices are unbounded).