This paper presents two fast cycle canceling algorithms for the submodular flow problem. The first uses an assignment problem whose optimal solution identifies most negative node-disjoint cycles in an auxiliary network. Canceling these cycles lexicographically makes it possible to obtain an optimal submodular flow in O(nh log(nC)) time, which almost matches the current fastest weakly polynomial...

We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a nondecreasing submodular function or minimize a nonincreasing supermodular function in the setting of bounded total curvature c. In the case of submodular maximization with curvature ...

We consider the problem of maximizing submodular functions; while this problem is known to be NP-hard, several numerically efficient local search techniques with approximation guarantees are available. In this paper, we propose a novel convex relaxation which is based on the relationship between submodular functions, entropies and probabilistic graphical models. In a graphical model, the entrop...

Submodular optimization has found many applications in machine learning andbeyond. We carry out the first systematic investigation of inference in probabilis-tic models defined through submodular functions, generalizing regular pairwiseMRFs and Determinantal Point Processes. In particular, we present L-FIELD, avariational approach to general log-submodular and log-supermodul...

We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds — a central problem in the study of network cascades. The majority of existing work on this problem, formally referred to as the influence maximization problem, is designed for submodular cascades. Despite the empirical evidence that many cascades are non-submodular, lit...

In this paper, we consider optimizing submodular functions that are drawn from some unknown distribution. This setting arises, e.g., in recommender systems, where the utility of a subset of items may depend on a user-specific submodular utility function. In modern applications, the ground set of items is often so large that even the widely used (lazy) greedy algorithm is not efficient enough. A...

Kiyohito NAGANO\dagger (11Lnnnnn!!!llll)))BggX) Abstract A linearly parameterized polymatroid intersection problem appears in the context of principal partitions. We consider a submodular intersection problem on a pair of strong-map sequences of submodular functions, which is an extension of the linearly parameterized polymatroid intersection problem to a nonlinearly parameterized one. We intro...

The notion of submodular partition functions generalizes many of well-known tree decompositions of graphs. For fixed k, there are polynomial-time algorithms to determine whether a graph has treewidth, branch-width, etc. at most k. Contrary to these results, we show that there is no sub-exponential algorithm for determining whether the width of a given submodular partition function is at most tw...

The notion of submodular partition functions generalizes many of well-known tree decompositions of graphs. For fixed k, there are polynomial-time algorithms to determine whether a graph has tree-width, branch-width, etc. at most k. Contrary to these results, we show that there is no sub-exponential algorithm for determining whether the width of a given submodular partition function is at most t...

The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank's discrete separation theorem for submodular/supermodular set functions and Edmon...