Submodular-function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximum-entropy sampling, and maximum facility-location problems. Our main result is that for any k ≥ 2 and any ε > 0, there is a natural local-search algorithm which has...

We propose a new optimization framework for summarization by generalizing the submodular framework of (Lin and Bilmes, 2011). In our framework the summarization desideratum is expressed as a sum of a submodular function and a nonsubmodular function, which we call dispersion; the latter uses inter-sentence dissimilarities in different ways in order to ensure non-redundancy of the summary. We con...

A general ordertheoretic linear programming model for the study of matroid-type greedy algorithms is introduced. The primal restrictions are given by so-called weakly increasing submodular functions on antichains. The LP-dual is solved by a Monge-type greedy algorithm. The model ooers a direct combinatorial explanation for many integrality results in discrete optimization. In particular, the su...

Submodular facility location functions are widely used for summarizing large datasets and have found applications ranging from sensor placement, image retrieval, and clustering. A significant problem is that evaluating such functions typically requires the calculation of pairwise benefits for all items, which is computationally unmanageable for large problems. In this paper we propose a sparsif...

We consider log-supermodular models on binary variables, which are probabilistic models with negative log-densities which are submodular. These models provide probabilistic interpretations of common combinatorial optimization tasks such as image segmentation. In this paper, we focus primarily on parameter estimation in the models from known upper-bounds on the intractable log-partition function...

The proximal problem for structured penalties obtained via convex relaxations of submodular functions is known to be equivalent to minimizing separable convex functions over the corresponding submodular polyhedra. In this paper, we reveal a comprehensive class of structured penalties for which penalties this problem can be solved via an efficiently solvable class of parametric maxflow optimizat...

Many large-scale machine learning problems – clustering, non-parametric learning, kernel machines, etc. – require selecting a small yet representative subset from a large dataset. Such problems can often be reduced to maximizing a submodular set function subject to various constraints. Classical approaches to submodular optimization require centralized access to the full dataset, which is impra...

We study two mixed robust/average-case submodular partitioning problems that we collectively call Submodular Partitioning. These problems generalize both purely robust instances of the problem (namely max-min submodular fair allocation (SFA) Golovin (2005) and min-max submodular load balancing (SLB) Svitkina and Fleischer (2008)) and also generalize average-case instances (that is the submodula...

Aproximity theorem is astatement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in acertain neighborhood of asolution to the relaxation. Proximity theorems have been used successfully, for example, in designing efficient algorithms for discrete resource allocation problems. After reviewing the recent results for $\mathrm{L}$-convex and...

Proof. We can establish the result for fixed x using the proof of Ohsaka and Yoshida. We have via taking c = L in their Lemma 4.4 that for any fixed x, |CVaRα(x) − ̂ CVaRα(x)|≤ with probability at least 1 − δ by taking s = Θ ( M2 2 log 1δ ) samples. Note that we cannot directly take union bound because the set of x ∈ P is not finite. Instead, we take a uniform grid of ( L1d )n points containing ...