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Modern Aspects of Submodularity

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Introduction to Discrete Convex Analysis

2012-03-19 , Murota, Kazuo

Discrete convex analysis is a theory that aims at a discrete analogue of convex analysis for nonlinear discrete optimization. Technically it is a nonlinear generalization of matroid/submodular function theory; matroids are generalized to M-convex functions and submodular set functions to L-convex function.This talk consists of two parts: In the first part fundamental concepts and theorems in discrete convex analysis are explained with reference to familiar combinatorial optimization problems like minimum spanning tree, shortest path, and matching. In view of the recent development in submodular function maximization, minimization and maximization algorithms in discrete convex analysis are explained in the second part of the talk. In particular, M-natural concave functions form a subclass of submodular functions that can be maximized in polynomial time.

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Submodular Function Optimization in Sensor and Social Networks

2012-03-19 , Krause, Andreas

Many applications in sensor and social networks involve discrete optimization problems. In recent years, it was discovered that many such problems have submodular structure. These problems include optimal sensor placement, informative path planning, active learning, influence maximization, online advertising and structure learning. In contrast to most previous approaches, submodularity allows to efficiently find provably (near-)optimal solutions. In this tutorial, I will give examples of submodular optimization problems arising in sensor and social networks, discuss algorithms for solving these problems and present results on real applications. I will also discuss recent work in online and adaptive optimization of submodular functions in these domains.

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Submodular Function Optimization in Sensor and Social Networks II

2012-03-19 , Krause, Andreas

Many applications in sensor and social networks involve discrete optimization problems. In recent years, it was discovered that many such problems have submodular structure. These problems include optimal sensor placement, informative path planning, active learning, influence maximization, online advertising and structure learning. In contrast to most previous approaches, submodularity allows to efficiently find provably (near-)optimal solutions. In this tutorial, I will give examples of submodular optimization problems arising in sensor and social networks, discuss algorithms for solving these problems and present results on real applications. I will also discuss recent work in online and adaptive optimization of submodular functions in these domains.

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Minimization and Maximization Algorithms in Discrete Convex Analysis

2012-03-19 , Murota, Kazuo

Discrete convex analysis is a theory that aims at a discrete analogue of convex analysis for nonlinear discrete optimization. Technically it is a nonlinear generalization of matroid/submodular function theory; matroids are generalized to M-convex functions and submodular set functions to L-convex function.This talk consists of two parts: In the first part fundamental concepts and theorems in discrete convex analysis are explained with reference to familiar combinatorial optimization problems like minimum spanning tree, shortest path, and matching. In view of the recent development in submodular function maximization, minimization and maximization algorithms in discrete convex analysis are explained in the second part of the talk. In particular, M-natural concave functions form a subclass of submodular functions that can be maximized in polynomial time.

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Optimization of Submodular Functions: Relaxations and Algorithms

2012-03-19 , Vondrak, Jan

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.

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