On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum-diameter edge addition problems

https://doi.org/10.1016/0167-6377(92)90007-PGet rights and content

Abstract

Given a graph G = (V, E), positive integer D < |V| and B,Minimum-Cardinality-Bounded-Diameter (MCBD) Edge Addition Problem is to find a superset of edges E′ ⊇ E such that the graph G′ = (V, E′) has diameter no greater than D and the total number of the new edges is minimized, while the Bounded-Cardinality-Minimum-Diameter (BCMD) Edge Addition Problem is to find a superset of edges E′ ⊇ E with |E′/E| ≤ B such that the diameter of G′ = (V, E′) is minimized. We prove that the MCBD case is NP-hard even when D = 2 and describe a polynomial heuristic for BCMD with a constant worst-case bound. We also show that finding a polynomial heuristic for MCBD with a constant worst-case bound is no easier than finding such a heuristic for the dominating set problem.

References (5)

There are more references available in the full text version of this article.

Cited by (68)

  • Augmenting graphs to minimize the radius

    2023, Computational Geometry: Theory and Applications
  • Almost optimal algorithms for diameter-optimally augmenting trees

    2022, Theoretical Computer Science
    Citation Excerpt :

    In the more general setting, the problem is NP-hard [7], not approximable within logarithmic factors unless P=NP [8], and some of its variants – parameterized w.r.t. the overall cost of added shortcuts and resulting diameter – are even W[2]-hard [9,10]. Therefore, several approximation algorithms have been developed for all these variations [8,11,12,9,13]. Finally, upper and lower bounds on the values of the diameters of the augmented graphs have also been investigated in [14–16].

  • A linear-time algorithm for radius-optimally augmenting paths in a metric space

    2021, Computational Geometry: Theory and Applications
View all citing articles on Scopus

Work supported in part by ONR contract N00014-90-J-1649, NSF contract DDM-8922712 and the Center for Telecommunications Research under NSF Grant CDR 84-21402.

View full text