Verification and sensitivity analysis of minimum spanning trees in linear time
SIAM Journal on Computing
An optimal minimum spanning tree algorithm
Journal of the ACM (JACM)
Sensitivity analysis of minimum spanning trees in sub-inverse-ackermann time
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Robust optimization in the presence of uncertainty
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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In classic optimization theory, the concept of stability refers to the study of how much and in which way the optimal solutions of a given minimization problem Π can vary as a function of small perturbations of the input data. Motivated by congestion problems arising in shortest-path based communication networks, in this paper we restrict ourselves to the case in which Π is actually a network design problem on a given graph G=(V,E,w) of |V|=n nodes, |E|=m edges, and with a positive real weight w(e) on each edge e∈E. We focus on a subclass of perturbations, that we call stretching perturbations, in which the weights of the edges of G can be increased by at most a fixed multiplicative real factor λ≥1. For this class of perturbations, we address the problem of computing the stability number of any given subgraph H of G containing at least an optimal solution of Π, namely the maximum stretching factor for which H keeps on maintaining an optimal solution. Furthermore, given a stretching factor λ, we study the problem of constructing a minimal subgraph of G with stability number greater or equal to λ. We develop a general technique to solve both problems. By applying this technique to the minimum spanning tree and the single-source shortest paths tree (SPT) problems, we obtain ${\cal O}(m\alpha(m,n))$ and ${\cal O}(mn(m+n \log n))$ time algorithms, respectively, where α(·,·) is the functional inverse of Ackermann's function. Furthermore, for the SPT problem, we show that if H coincides with the set of all optimal solutions, then the time complexity can be reduced to ${\cal O}(mn)$. Finally, for the single-source single-destination shortest path problem, if the optimal solutions of the input instance happen to form a set of vertex-disjoint paths, and H coincides with this set, then we show that we can compute the stability number in ${\cal O}(mn + n^2 \log n)$ time.