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

Published in Elsevier

2015

Volume: 33

Pages: 71 - 80

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 NP-complete problems and they have been studied from the parameterized complexity point of view in [1]. Here, we prove that for any ε>0, both the problems cannot be approximated within a factor (1-ε)log n, unless NP⊆Dtime(nlog log n). We also show that for any ε>0, MDD(min) cannot be approximated within a factor (1-ε)log n on bipartite graphs, unless NP⊆Dtime(nlog log n), and that for any ε>0, MDD(max) cannot be approximated within a factor (1/2-ε)log n on bipartite graphs, unless NP⊆Dtime(nlog log 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(log n), 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. © 2015 Elsevier B.V.

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About the journal

Journal | Data powered by TypesetJournal of Discrete Algorithms |
---|---|

Publisher | Data powered by TypesetElsevier |

ISSN | 15708667 |

Open Access | Yes |

Concepts (13)

- Approximation algorithms
- Computational complexity
- Graphic methods
- Polynomial approximation
- Bipartite graphs
- CONSTANT DEGREE
- FACTOR APPROXIMATION ALGORITHMS
- Hardness of approximation
- NODE DELETION
- PARAMETERIZED COMPLEXITY
- Polynomial-time
- UNWEIGHTED GRAPHS
- Graph theory