Upgrading trees under diameter and budget constraints

Given a tree T (V, E) endowed with a length function l and a cost function c, the diameter lowering problem consists in finding the reals 0 ≤ x(e) ≤ l(e), e E such that the tree obtained from T by decreasing the length of every edge e by x(e) units has a minimal diameter subject to the constraint ¥e Ec(e)x(e) ≤ B, where B is the available budget (analogously, one can minimize the cost of lowering subject to a diameter constraint). We present an O( V ) algorithm for solving this problem by developing and using algorithms of similar complexity for related eccentricity lowering problems. © 2002 Wiley Periodicals, Inc.