Solving stochastic optimization problems under partial observability, where one needs to adaptively make decisions with uncertain outcomes, is a fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. We prove that if a problem satisfies this property, a simple adaptive greedy...

The supermodular covering knapsack set is the discrete upper level set of a non-decreasing supermodular function. Submodular and supermodular knapsack sets arise naturally when modeling utilities, risk and probabilistic constraints on discrete variables. In a recent paper Atamtürk and Narayanan [6] study the lower level set of a non-decreasing submodular function. In this complementary paper we...

A submodular polyhedron is a polyhedron associated with a submodular function. This paper presents a strongly polynomial time algorithm for line search in submodular polyhedra with the aid of a fully combinatorial algorithm for submodular function minimization. The algorithm is based on the parametric search method proposed by Megiddo.

In this work we show, that for each permuted submodular MinSum problem (Energy Minimization Task) the corresponding submodular MinSum problem can be found in polynomial time. It follows, that permuted submodular MinSum problems are exactly solvable by transforming them into corresponding submodular tasks followed by applying standart approaches (e.g. using MinCutMaxFlow based techniques).

SUMMARY We first describe fundamental results about submodular functions and submodular flows, which lay a basis for devising efficient algorithms for submodular flows. We then give a comprehensive survey on algorithms for submodular flows and show some possible future research directions.

Submodular function minimization is a fundamental optimization problem that arises in several applications in machine learning and computer vision. The problem is known to be solvable in polynomial time, but general purpose algorithms have high running times and are unsuitable for large-scale problems. Recent work have used convex optimization techniques to obtain very practical algorithms for ...

In this paper we model the multi-level uncapacitated facility location problem as two different combinatorial optimization problems. The first one uses a set of vertices as combinatorial objects to represent solutions whereas the second one uses a set of paths. An interesting observation is that the real-valued set function associated with the first combinatorial problem does not satisfy the su...

This paper considers the design and effective interfaces of a distributed robotic formation running planar weighted bearing-compass dynamics. We present results which support methodologies to construct formation topologies using submodular optimization techniques. Further, a convex optimization framework is developed for the selection of edge weights which increase performance. We explore a met...

Determinantal point processes (DPPs) have recently been proposed as computationally efficient probabilistic models of diverse sets for a variety of applications, including document summarization, image search, and pose estimation. Many DPP inference operations, including normalization and sampling, are tractable; however, finding the most likely configuration (MAP), which is often required in p...