Fast reoptimization for the minimum spanning tree problem

• Authors: Nicolas Boria;Vangelis Th. Paschos
• Venue: Journal of Discrete Algorithms
• Year: 2010

We study reoptimization versions of the minimum spanning tree problem. The reoptimization setting can generally be formulated as follows: given an instance of the problem for which we already know some optimal solution, and given some ''small'' perturbations on this instance, is it possible to compute a new (optimal or at least near-optimal) solution for the modified instance without ex nihilo computation? We focus on two kinds of modifications: node-insertions and node-deletions. When k new nodes are inserted together with their incident edges, we mainly propose a fast strategy with complexity O(kn) which provides a max{2,3-(2/(k-1))}-approximation ratio, in complete metric graphs and another one that is optimal with complexity O(nlogn). On the other hand, when k nodes are deleted, we devise a strategy which in O(n) achieves approximation ratio bounded above by 2@?|L"m"a"x|/2@? in complete metric graphs, where L"m"a"x is the longest deleted path and |L"m"a"x| is the number of its edges. For any of the approximation strategies, we also provide lower bounds on their approximation ratios.