Integral packing of trees and branchings

Abstract

This article continues the discussion of the author`s results on strictly polynomial algorithms for network strength problems (it is assumed that the reader is familiar with the previous publications). It considers the problem of optimal integral packing of spanning trees in a graph and proposes a strictly polynomial algorithm for the solution of this problem. The spanning tree packing and network covering algorithms described produce noninteger solutions. However, the Tutte-Nash-Williams theorem provides a good characterization for the solution of the corresponding problems for trees with integral cardinalities. Interger solutions can be obtained by Cunningham`s general algorithm, which produces an integer solution for the problem of packing of bases of a polymatroid polyhedron. This algorithm, however, is characterized by high time complexity. Moreover, the number of packed bases (in our case, spanning trees) in Seriver`s modification is double the theoretical minimum. In this paper, we apply the results to propose on O(n{sup 2} mp) algorithm for the problem of integral packing of spanning trees, where n and m respectively are the number of vertices and edges in the graph G and p is the time complexity of the maximum flow problem on G. The algorithm constructs a basis solution, so thatmore » the optimal solution contains a minimum number of spanning trees of nonzero cardinalities. In other words, the number of nonzero components forming the optimal packing does not exceed n. The proposed algorithm is easily modified for the solution of problems of minimum integral packing and covering described elswhere, and its elaboration for the present case is left to the reader. The spanning tree packing problem is transformed into a similar problem for digraphs, specifically, the problem of packing branchings into a given digraph with a distinguished root. A good characterization of this problem is provided by the Edmonds theorem.« less