28,125 research outputs found

    On the approximability of robust spanning tree problems

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    In this paper the minimum spanning tree problem with uncertain edge costs is discussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within O(log1ϵn)O(\log^{1-\epsilon} n) for any ϵ>0\epsilon>0 unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within O(logn)O(\log n) unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance ratio of O(log2n)O(\log^2 n) for min-max and 2-stage min-max problems are also proposed

    An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles

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    We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurobi

    Connection Situations under Uncertainty

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    cooperative cost games;minimum cost spanning tree situations;robustness;worst-case scenario;input interval data;uncertainty
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