We show that Karger's randomized contraction method (SODA 93) can be adapted to multiobjective global minimum cut problems with a constant number of edge or node budget constraints to give efficient algorithms. For global minimum cuts with a single edge-budget constraint, our extension of the randomized contraction method has running time tilde{O}(n^3) in an n-node graph improving upon the best-known randomized algorithm with running time tilde{O}(n^4) due to Armon and Zwick (Algorithmica 2006). Our analysis also gives a new upper bound of O(n^3) for the number of optimal solutions for a single edge-budget min cut problem. For the case of (k-1) edge-budget constraints, the extension of our algorithm saves a logarithmic factor from the best-known randomized running time of O(n^{2k} log^3 n). A main feature of our algorithms is to adaptively choose, at each step, the appropriate cost function used in the random selection of edges to be contracted. For the global min cut problem with a constant number of node budgets, we give a randomized algorithm with running time tilde{O}(n^2), improving the current best determinisitic running time of O(n^3) due to Goemans and Soto (SIAM Journal on Discrete Mathematics 2013). Our method also shows that the total number of distinct optimal solutions is bounded by O(n^2) as in the case of global min-cuts. Our algorithm extends to the node-budget constrained global min cut problem excluding a given sink with the same running time and bound on number of optimal solutions, again improving upon the best-known running time by a factor of O(n). For node-budget constrained problems, our improvements arise from incorporating the idea of merging any infeasible super-nodes that arise during the random contraction process. In contrast to cuts excluding a sink, we note that the node-cardinality constrained min-cut problem containing a given source is strongly NP-hard using a reduction from graph bisection.

This paper deals with the problem of capacity expansion of a network under independent uncertain demands defined by interval sets. In this context, decisions about capacity expansion must be made before the demands are revealed. Standard robust models require the definition of an uncertainty domain and look for the minimum cost solution able to satisfy any demand within this domain. We propose, justify, and illustrate an alternative robust model based on a bi‐objective formulation. Therefore, in addition to the cost criterion, we consider a second criterion, related to the Quality of Service, which measures the ability of a solution to handle any demand. The decision‐maker can be interested in efficient solutions offering a compromise between these criteria. We study the complexity of the enumeration of the corresponding nondominated set, and propose exact and approximation algorithms. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 185–199 2016