Balancing problems in acyclic networks

A directed acyclic network with nonnegative integer arc lengths is called balanced if any two paths with common endpoints have equal lengths. In the buffer assignment problem such a network is given, and the goal is to balance it by increasing arc lengths by integer amounts (called buffers), so that the sum of the amounts added is minimal. This problem arises in VLSI design, and was recently shown to be polynomial for rooted networks. Here we give simple procedures which solve several generalizations of this problem in strongly polynomial time, using ideas from network flow theory. In particular, we solve a weighted version of the problem, extend the results to nonrooted networks, and allow upper bounds on buffers. We also give a strongly polynomial algorithm for solving the min-max buffer assignment problem, based on a strong proximity result between fractional and integer balanced solutions. Finally, we show that the problem of balancing a network while minimizing the number of arcs with positive buffers is NP-hard.