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 pβV, MDD(min) is the problem of finding a minimum
weight vertex set SβVβ{p} such that p becomes the
minimum degree vertex in G[VβS]; and MDD(max) is the problem of
finding a minimum weight vertex set SβVβ{p} such that
p becomes the maximum degree vertex in G[VβS]. These are known
NP-complete problems and have been studied from the parameterized complexity
point of view in previous work. Here, we prove that for any Ο΅>0,
both the problems cannot be approximated within a factor (1βΟ΅)logn, unless NPβDTIME(nloglogn). We also show that for any
Ο΅>0, MDD(min) cannot be approximated within a factor (1βΟ΅)logn on bipartite graphs, unless NPβDTIME(nloglogn), and that for any Ο΅>0, MDD(max) cannot be approximated within a
factor (1/2βΟ΅)logn on bipartite graphs, unless NPβDTIME(nloglogn). We give an O(logn) factor approximation algorithm
for MDD(max) on general graphs, provided the degree of p is O(logn). We
then show that if the degree of p is nβO(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.Comment: 16 pages, 4 figures, submitted to Elsevier's Journal of Discrete
Algorithm