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In this talk, we will outline a polynomial time algorithm for solving the problem of partial covering of trees with n 1 balls of radius R 1 and n 2 balls of radius R 2 ( R 1 < R 2 ) so as to maximize the total number of covered vertices. We will then show that the solutions provided by this algorithm in the particular case R 1 = R − 1 , R 2 = 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 = 2 R + 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.

Mixed Covering of Trees and the Augmentation Problem with Odd Diameter Constraints