Clustering with submodular functions has been of interest over the last few years. Symmetric submodular functions are of particular interest as minimizing them is significantly more efficient and they include many commonly used functions in practice viz. graph cuts, mutual information. In this paper we propose a novel constraint to make clustering actionable which is motivated by applications a...

This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions: For a submodular integrally convex function g(p1, . . . , pn), the function g̃ defined by g̃(p0, p1, . . . , pn) = g(p1 − p0, . . . , pn − p0) is an L-convex function, and vice versa. This fact implies, in combination with known results for L-convex functions, tha...

We consider the problem of maximizing a nonnegative submodular set function f : 2N → R+ subject to a p-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are non-monotone. We describe deterministic and randomized algor...

A k-submodular function is a generalization of a submodular function, where the input consists of k disjoint subsets, instead of a single subset, of the domain. Many machine learning problems, including influence maximization with k kinds of topics and sensor placement with k kinds of sensors, can be naturally modeled as the problem of maximizing monotone k-submodular functions. In this paper, ...

Krause et al. [2008] has an indirect proof that F ′ is not submodular. They give an example that shows the greedy algorithm can do arbitrarily bad when maximizing F ′ under a cardinality constraint. If F ′ were submodular the greedy algorithm would give a (1 − 1/e) approximation for maximizing under a cardinality constraint. We give a direct proof that F ′ is not submodular for completeness. In...

Submodular functions are common in combinatorics; examples include the cut capacity function of a graph and the rank function of a matroid. The submodular function minimization problem generalizes the classical minimum cut problem and also contains a number of other combinatorial optimization problems as special cases. In this thesis, we study submodular function minimization and two related pr...

Using polarity, we give an outer polyhedral approximation for the epigraph of set functions. For a submodular function, prove that corresponding polar relaxation is exact; hence, it equivalent to Lovász extension. The approach provides alternative proof convex hull description function. Computational experiments show inequalities from approximations can be effective as cutting planes solving we...

We investigate the problem of maximizing sum submodular and supermodular functions under a fairness constraint. This function is non-submodular in general. For an offline model, we introduce two approximation algorithms: A greedy algorithm threshold algorithm. streaming propose one-pass also analyze ratios these algorithms, which all depend on total curvature function. The computable polynomial...

We consider an online prediction problem of combinatorial concepts where each combinatorial concept is represented as a vertex of a polyhedron described by a submodular function (base polyhedron). In general, there are exponentially many vertices in the base polyhedron. We propose polynomial time algorithms with regret bounds. In particular, for cardinality-based submodular functions, we give O...

The simplex algorithm for linear programming is based on the fact that any local optimum with respect to the polyhedral neighborhood is also a global optimum. We show that a similar result carries over to submodular maximization. In particular, every local optimum of a constrained monotone submodular maximization problem yields a 1/2-approximation, and we also present an appropriate extension t...