1 Special cases of submodular flows We saw last time that orientation of a 2k-edge connected graph into a k-arc connected digraph and the Lucchesi and Younger Theorem were special cases of submodular flows. Other familiar problems can also be phrased as submodular flows. Let C = 2 V \ {∅, V } and let f be identically zero. Then for any U ∈ C, x(δ in (U)) − x(δ out (U) ≤ 0 (1) x(δ in (V \ U)) − ...

Motivated by extremely large-scale machine learning problems, we introduce a new multistage algorithmic framework for submodular maximization (called MultGreed), where at each stage we apply an approximate greedy procedure to maximize surrogate submodular functions. The surrogates serve as proxies for a target submodular function but require less memory and are easy to evaluate. We theoreticall...

We propose and study a new class of submodular functions called deep submodular functions (DSFs). We define DSFs and situate them within the broader context of classes of submodular functions in relationship both to various matroid ranks and sums of concave composed with modular functions (SCMs). Notably, we find that DSFs constitute a strictly broader class than SCMs, thus motivating their use...

This paper introduces a general matroid-theoretic construction which includes, as special cases, elementary lifts of matroids and bias matroids of biased graphs. To perform the construction on a matroid A/ , it is necessary (but not sufficient) to have a submodular function inducing M . Elementary lifts are obtained when the submodular function chosen is the rank function of M . We define what ...

We investigate the performance of the GREEDY algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of GREEDY for maximizing submodular functions, there are few guarantees for non-submodular ones. However, GREEDY enjoys strong empirical performance for many important non-submodular functi...

We consider the problem of characterizing the minimum of a submodular function when the minimization is restricted to a family of subsets. We show that, for many interesting cases, there exist two elements a and b of the groundset such that the problem is equivalent to the problem of minimizing the submodular function over the sets containing a but not b. This leads to a polynomial-time algorit...

We use the property of unimodular functions to perform approximate inference with dual decomposition in binary labeled graphs. Exact inference is possible for a subclass of binary labeled graphs that have unimodular functions. We call such graphs unimodular graphs. These are graphs where the submodular and nonsubmodular edges follow a specific pattern– essentially that an isomorphism or ”flippi...

Submodular functions arewell-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call weakly submodular functions. Our extension is somewhat unusual in that it includes some (mildly) supermodular functions. We show that several natura...

We consider the problem of finding M best diverse solutions of energy minimization problems for graphical models. Contrary to the sequential method of Batra et al., which greedily finds one solution after another, we infer all M solutions jointly. It was shown recently that such jointly inferred labelings not only have smaller total energy but also qualitatively outperform the sequentially obta...

In this paper, we consider the problem of producing balanced clusterings with respect to a submodular objective function. Submodular objective functions occur frequently in many applications, and hence this problem is broadly applicable. We show that the results of Patkar and Narayanan [8] can be applied to cases when the submodular function is derived from a bipartite object-feature graph, and...