We consider the problem of jointly inferring the M -best diverse labelings for a binary (high-order) submodular energy of a graphical model. Recently, it was shown that this problem can be solved to a global optimum, for many practically interesting diversity measures. It was noted that the labelings are, so-called, nested. This nestedness property also holds for labelings of a class of paramet...

Let G = (V, E) be a graph and w: E -» R+ be a length function. Given S C V, a Steiner tour is a cycle passing at least once through each vertex of S . In this paper we investigate naturally submodular graphs: graphs for which the length function of the Steiner tours is submodular. We provide two characterizations of naturally submodular graphs, an 0(n) time algorithm for identifying such graphs...

A probabilistic graphical model encodes conditional independences among random variables, which is related to factorisable distributions. Moreover, the entropy of a probability distribution on a set of discrete random variables is always bounded by the entropy of its factorisable counterpart. This is due to the submodularity of entropy on the set of discrete random variables. Submodular functio...

A set function f defined on the subsets of a finite set V is said to be submodular if it satisfies f(X) + f(Y ) ≥ f(X ∪ Y ) + f(X ∩ Y ), ∀X, Y ⊆ V. Submodular functions are discrete analogues of convex functions. They arise in various branches of applied mathematics such as game theory, information theory, and queueing theory. Examples include the matroid rank functions, the cut capacity functi...

We introduce submodular hypergraphs, a family of hypergraphs that have different submodular weights associated with different cuts of hyperedges. Submodular hypergraphs arise in clustering applications in which higher-order structures carry relevant information. For such hypergraphs, we define the notion of p-Laplacians and derive corresponding nodal domain theorems and k-way Cheeger inequaliti...

Submodularity (or concavity) is considered as an important property in the field of cooperative game theory. In this article, we characterize submodular minimum coloring games and submodular minimum vertex cover games. These characterizations immediately show that it can be decided in polynomial time that the minimum coloring game or the minimum vertex cover game on a given graph is submodular ...

We study the problem of optimal flexibility capacity portfolio selection by introducing a new notion of submodularity in correspondences, which extends the classical notion of submodular functions. In particular, we prove that the correspondence that maps flexible resources to the set of demands that they can process is submodular, and use the properties of submodular correspondences to compare...

Submodular ow problems, introduced by Edmonds and Giles 2], generalize network ow problems. Many algorithms for solving network ow problems have been generalized to submodular ow problems (cf. references in Fujishige 4]), e.g. the cycle canceling method of Klein 9]. For network ow problems, the choice of minimum-mean cycles in Goldberg and Tarjan 6], and the choice of minimum-ratio cycles in Wa...

Submodular continuous functions are a category of (generally) non-convex/nonconcave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation guarantees. Specifically, I) for monotone submodular continuous functions with an additional diminishing returns property, we propose a Frank-Wolfe style algor...

An M-convex function is a nonlinear discrete function defined on integer points introduced by Murota in 1996, and the M-convex submodular flow problem is one of the most general frameworks of efficiently solvable combinatorial optimization problems. It includes the minimum cost flow and the submodular flow problems as its special cases. In this paper, we first devise a successive shortest path ...