Minimizing broadcast costs under edge reductions in tree networks

We study the broadcasting of messages in tree networks under edge reductions. When an edge is reduced, its cost becomes zero. Edge reductions model the decrease or elimination of broadcasting costs between adjacent nodes in the network. Let T be an n-vertex tree and B be a target broadcast cost. We present an O(n)-time algorithm for determining the minimum number of edges of T to reduce so that a broadcast cost of B can be achieved. We present an O(n log n)-time algorithm to determine the minimum number of edges to reduce so that a broadcast initiated at an arbitrary vertex of T costs at most B. Characterizations of where edge reductions are placed underlie both algorithms and imply that reduced edges can be centrally located.