On the complexity of making a distinguished vertex minimum or maximum degree by vertex deletion

In this paper, we investigate the approximability of two node deletion problems. Given a vertex weighted graph G = (V,E) and a specified, or “distinguished” vertex p ∈ V , MDD(min) is the problem of finding a minimum weight vertex set S ⊆ V \ {p} such that p becomes the minimum degree vertex in G[V \ S]; and MDD(max) is the problem of finding a minimum weight vertex set S ⊆ V \{p} such that p becomes the maximum degree vertex in G[V \ S]. These are known NPcomplete problems and have been studied from the parameterized complexity point of view in [1]. Here, we prove that for any ǫ > 0, both the problems cannot be approximated within a factor (1− ǫ) logn, unless NP ⊆ Dtime(n ). We also show that for any ǫ > 0, MDD(min) cannot be approximated within a factor (1 − ǫ) logn on bipartite graphs, unless NP ⊆ Dtime(n ), and that for any ǫ > 0, MDD(max) cannot be approximated within a factor (1/2 − ǫ) logn on bipartite graphs, unless NP ⊆ Dtime(n ). We give an O(log n) factor approximation algorithm for MDD(max) on general graphs, provided the degree of p is O(log n). We then show that if the degree of p is n−O(logn), a similar result holds for MDD(min). We prove that MDD(max) is APX-complete on 3-regular unweighted graphs and provide an approximation algorithm with ratio 1.583 when G is a 3-regular unweighted graph. In addition, we show that MDD(min) can be solved in polynomial time when G is a regular graph of constant degree.