In this paper we present a polynomial time algorithm for solving the problem of partial covering of trees with n1 balls of radius R1 and n2 balls of radius R2 (R1 < R2) to maximize the total number of covered vertices. The solutions provided by this algorithm in the particular case R1 = R – 1, R2 = R can be used to obtain for any integer δ > 0 a factor (2+1/δ) approximation algorithm for solving the following augmentation problem with odd diameter constraints D = 2R + 1: Given a tree T, add a minimum number of new edges such that the augmented graph has diameter ≤ D. The previous approximation algorithm of Ishii, Yamamoto, and Nagamochi (2003) has factor 8.
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Chepoi, V., Estellon, B., Nouioua, K. et al. Mixed Covering of Trees and the Augmentation Problem with Odd Diameter Constraints. Algorithmica 45, 209–226 (2006). https://doi.org/10.1007/s00453-005-1183-9
- Partial covering
- Augmentation problem
- Dynamical programming
- Approximation algorithms