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In this work, we consider the problem of maximizing the reliability of connections in mesh networks against failure scenarios in which multiple links may fail simultaneously. We consider the single-path connection problem as well as multiple-path (protected) connection problems. The problems are formulated as minimum-color path problems, where each link is associated with one or more colors, and each color corresponds to a given failure event Thus, when a certain color fails, all links which include that color will fail. In a single-path problem, by minimizing the number of colors on the path, the failure probability of the path can be minimized if all colors have the same probability of causing failures. In the case of two paths, where one path is a protection path, if all colors have the same probability of causing failures, the problem becomes that of finding two link-disjoint paths which either have a minimum total number of colors, or which have a minimum number of overlapping colors. By minimizing the total number of colors, the probability that a failure will occur on either of the paths is minimized. On the other hand, by minimizing the number of overlapping colors, the probability that a single failure event will cause both paths to fail simultaneously is minimized. The problems are proved to be NP-complete, and ILP formulations are developed. Heuristic algorithms are proposed for larger instances of the problems, and the heuristics are evaluated through simulation.