Optimization of Submodular Functions: Relaxations and Algorithms
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In this lecture, we cover the basic variants of optimization problems involving submodular functions, and known algorithms for solving them. Some particular problems that will be mentioned: the unconstrained minimization/ maximization of a submodular function, submodular optimization subject to packing/covering constraints, and welfare maximization in combinatorial auctions. Starting from greedy and local search algorithms, we will move on to continuous relaxations of these problems and their role in deriving efficient algorithms. Two important concepts that we will cover here are the Lovasz extension (for minimization), and the multilinear extension (for maximization) of submodular functions.