Approximation Algorithms for Generalized Bounded Tree Cover.

A tree cover is a collection of subtrees of a graph such that each vertex is a part of at least one subtree. The bounded tree cover problem (BTC) requires to find a tree cover with minimum number of subtrees of bounded weight. This paper considers two generalized versions of BTC. The first problem deals with graphs having multiple weight functions. Two variations called strong and weak tree cover problems are defined. In strong tree cover, every subtree must be bounded with respect to all weight functions, whereas in weak tree cover, each subtree must be bounded with respect to at least one weight function. We consider only metric weight functions. A 4-approximation algorithm for strong tree cover and an (O(log n))-approximation algorithm for weak tree cover problem has been proposed. In the second problem, the objective is to find a tree cover where the bounds of the subtrees are not necessarily same. We show that this problem cannot be approximated within a constant factor unless P=NP.