In this paper, we shall study a unified framework of robust submodular optimization. We study this problem both from a minimization and maximization perspective (previous work has only focused on variants of robust submodular maximization). We do this under a broad range of combinatorial constraints including cardinality, knapsack, matroid as well as graph based constraints such as cuts, paths, matchings and trees. Furthermore, we also study robust submodular minimization and maximization under multiple submodular upper and lower bound constraints. We show that all these problems are motivated by important machine learning applications including robust data subset selection, robust co-operative cuts and robust co-operative matchings. In each case, we provide scalable approximation algorithms and also study hardness bounds. Finally, we empirically demonstrate the utility of our algorithms on real world applications.
 In this paper the minmax (regret) versions of some basic polynomially solvable deterministic network problems are discussed. It is shown that if the number of scenarios is unbounded, then the problems under consideration are not approximable within log^1^-^@eK for any @e>0 unless NP@?DTIME(n^p^o^l^y^l^o^g^n), where K is the number of scenarios. In this paper the shortest path and the minimum spanning tree problems in a graph with @math nodes and @math cost scenarios (objectives) are discussed. In order to choose a solution the min-max criterion is applied. The minmax versions of both problems are hard to approximate within @math for any @math . The best approximation algorithm for the min-max shortest path problem, known to date, has approximation ratio of @math . On the other hand, for the min-max spanning tree, there is a randomized algorithm with approximation ratio of @math . In this paper a deterministic @math -approximation algorithm for min-max shortest path is constructed. For min-max spanning tree a deterministic @math -approximation algorithm is proposed, which works for a large class of graphs and a randomized @math -approximation algorithm, which can be applied to all graphs, is constructed. It is also shown that the approximation ratios obtained are close to the integrality gaps of the corresponding LP relaxations. 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(log^1^-^@en) for any @e>0 unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within O(logn) unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance bound of O(log^2n) for min-max and 2-stage min-max problems are also proposed. Min-max and min-max regret criteria are commonly used to define robust solutions. After motivating the use of these criteria, we present general results. Then, we survey complexity results for the min-max and min-max regret versions of some combinatorial optimization problems: shortest path, spanning tree, assignment, min cut, min s-t cut, knapsack. Since most of these problems are NP-hard, we also investigate the approximability of these problems. Furthermore, we present algorithms to solve these problems to optimality.  While the complexity of min-max and min-max regret versions of most classical combinatorial optimization problems has been thoroughly investigated, there are very few studies about their approximation. For a bounded number of scenarios, we establish general approximation schemes which can be used for min-max and min-max regret versions of some polynomial or pseudo-polynomial problems. Applying these schemes to shortest path, minimum spanning tree, minimum weighted perfect matching on planar graphs, and knapsack problems, we obtain fully polynomial-time approximation schemes with better running times than the ones previously presented in the literature. 