Reflection methods for user-friendly submodular optimization

Recently, it has become evident that submodularity naturally captures widely occurring concepts in machine learning, signal processing and computer vision. Consequently, there is need for efficient optimization procedures for submodular functions, especially for minimization problems. While general submodular minimization is challenging, we propose a new method that exploits existing decomposability of submodular functions. In contrast to previous approaches, our method is neither approximate, nor impractical, nor does it need any cumbersome parameter tuning. Moreover, it is easy to implement and parallelize. A key component of our method is a formulation of the discrete submodular minimization problem as a continuous best approximation problem that is solved through a sequence of reflections, and its solution can be easily thresholded to obtain an optimal discrete solution. This method solves both the continuous and discrete formulations of the problem, and therefore has applications in learning, inference, and reconstruction. In our experiments, we illustrate the benefits of our method on two image segmentation tasks.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013

Convex Analysis and Optimization with Submodular Functions: a Tutorial

Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of submodular functions is presented, in a self-contained way, with all results shown from first principles. A good knowledge of convex analysis is assumed

Submodular linear programs on forests

A general linear programming model for an order-theoretic analysis of both Edmonds' greedy algorithm for matroids and the NW-corner rule for transportation problems with Monge costs is introduced. This approach includes the model of Queyranne, Spieksma and Tardella (1993) as a special case. We solve the problem by optimal greedy algorithms for rooted forests as underlying structures. Other solvable cases are also discussed

Thresholded Covering Algorithms for Robust and Max-Min Optimization

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".Comment: 24 page

Minimum Cost Multicast with Decentralized Sources

In this paper we study the multisource multicast problem where every sink in a given directed acyclic graph is a client and is interested in a common file. We consider the case where each node can have partial knowledge about the file as a side information. Assuming that nodes can communicate over the capacity constrained links of the graph, the goal is for each client to gain access to the file, while minimizing some linear cost function of number of bits transmitted in the network. We consider three types of side-information settings:(ii) side information in the form of linearly correlated packets; and (iii) the general setting where the side information at the nodes have an arbitrary (i.i.d.) correlation structure. In this work we 1) provide a polynomial time feasibility test, i.e., whether or not all the clients can recover the file, and 2) we provide a polynomial-time algorithm that finds the optimal rate allocation among the links of the graph, and then determines an explicit transmission scheme for cases (i) and (ii)