k-submodular functions, introduced by Huber and Kolmogorov, are functions defined on {0, 1, 2, . . . , k}n satisfying certain submodular-type inequalities. k-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by k-submodular functions play key roles in design of efficient, approximation, or FPT algorithms. Motivated by this, we consider the following pr...

Building on recent results for submodular minimization with combinatorial constraints, and on online submodular minimization, we address online approximation algorithms for submodular minimization with combinatorial constraints. We discuss two types of algorithms and outline approximation algorithms that integrate into those.

A function f : ZE+ → R+ is DR-submodular if it satisfies f(x+χi)−f(x) ≥ f(y+χi)−f(y) for all x ≤ y, i ∈ E. Recently, the problem of maximizing a DR-submodular function f : ZE+ → R+ subject to a budget constraint ‖x‖1 ≤ B as well as additional constraints has received significant attention [6, 7, 5, 8]. In this note, we give a generic reduction from the DR-submodular setting to the submodular se...

We introduce a method to learn a mixture of submodular “shells” in a large-margin setting. A submodular shell is an abstract submodular function that can be instantiated with a ground set and a set of parameters to produce a submodular function. A mixture of such shells can then also be so instantiated to produce a more complex submodular function. What our algorithm learns are the mixture weig...

Interactive submodular set cover is an interactive variant of submodular set cover over a hypothesis class of submodular functions, where the goal is to satisfy all sufficiently plausible submodular functions to a target threshold using as few (cost-weighted) actions as possible. It models settings where there is uncertainty regarding which submodular function to optimize. In this paper, we pro...

We introduce a class of discrete divergences on sets (equivalently binary vectors)that we call the submodular-Bregman divergences. We consider two kinds ofsubmodular Bregman divergence, defined either from tight modular upper or tightmodular lower bounds of a submodular function. We show that the properties ofthese divergences are analogous to the (standard continuous) Bregman d...

The seminal work by Edmonds [9] and Lovász [39] shows the strong connection between submodular functions and convex functions. Submodular functions have tight modular lower bounds, and a subdifferential structure [16] in a manner akin to convex functions. They also admit polynomial time algorithms for minimization and satisfy the Fenchel duality theorem [18] and the Discrete Seperation Theorem ...

A k-submodular function is an extension of a submodular function in that its input is given by k disjoint subsets instead of a single subset. For unconstrained nonnegative ksubmodular maximization, Ward and Živný proposed a constant-factor approximation algorithm, which was improved by the recent work of Iwata, Tanigawa and Yoshida presenting a 1/2-approximation algorithm. Iwata et al. also pro...

In this paper, we consider the problem of constrained maximization of the minimum of a set of submodular functions, in which the goal is to find solutions that are robust to worst-case values of the objective functions. Unfortunately, this problem is both non-submodular and inapproximable. In the case where the submodular functions are monotone, an approximate solution can be found by relaxing ...