In a lot of numerical computations, we can ask about the "stability" or "robustness" of the answer. This kind of question can be asked for combinatorial problems as well; here's one way of phrasing the question for the Minimum Spanning Tree Problem.
Suppose you are given a graph G = (V, E), with a cost ce on each edge e. We view the costs as quantities that have been measured experimentally, subject to possible errors in measurement. Thus, the minimum spanning tree one computes for G may not in fact be the "real" minimum spanning tree.
Given error parameters e> 0 and k > 0, and a specific edge e' = (u, v), you would like to be able to make a claim of the following form.
(*) Even if the cost of each edge were to be changed by at most e (either increased or decreased), and the costs of k of the edges other than e' were further changed to arbitrarily different values, the edge e' would still not belong to any minimum spanning tree of G.
Such a property provides a type of guarantee that e' is not likely to belong to the minimum spanning tree, even assuming significant measurement error.
Give a polynomial-time algorithm that takes G, e', e, and k, and decides whether or not property (*) holds for e'.
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