Generalized Maximum Independent Sets for Trees in Subquadratic Time

In this paper we consider a special case of the Maximum Weighted Independent Set problem for graphs: given a vertex- and edge-weighted tree Τ = (V, E) where |V| = n, and a real number b, determine the largest weighted subset P of V such that the distance between the two closest elements of P is at least b. We present an O(n log3 n) algorithm for this problem when the vertices have unequal weights. The space requirement is O(n log n). This is the first known subquadratic algorithm for the problem. This solution leads to an O(n log4 n) algorithm to the previously-studied Weighted Max-Min Dispersion Problem.