Two seemingly unrelated problems, scheduling a multiclass queueing system and minimizing a submodular function, share a rather deep connection via the polymatroid that is characterized by a submodular set function on the one hand and represents the performance polytope of the queueing system on the other hand. We first develop what we call a grouping algorithm that solves the queueing schedulin...

This paper focuses on sensor scheduling for state estimation, which consists of a network of noisy sensors and a discrete-time linear system with process noise. As an energy constraint, only a subset of sensors can take a measurement at each time step. These measurements are fused into a common state estimate using a Kalman filter and the goal is to schedule the sensors to minimize the estimati...

In this thesis we consider two combinatorial optimization problems that relate to the field of persistent monitoring. In the first part, we extend the classic problem of finding the maximum weight Hamiltonian cycle in a graph to the case where the objective is a submodular function of the edges. We consider a greedy algorithm and a 2-matching based algorithm, and we show that they have approxim...

We present a branch and bound method for maximizing an arbitrary set function V : 2 θ → R . By decomposing θ as f-δ , where f is a submodular function and δ is the cut function of a (simple, undirected) graph G with vertex set V, our original problem is reduced to a sequence of submodular maximization problems. We characterize a class of submodular functions, which when maximized in the subprob...

It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive p...

The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0-1 knapsack set. One motivation for studying the submodular knapsack polytope is to address 0-1 programming problems with uncertain coefficients. Under various assumptions, a probabilistic constraint on 0-1 variables can be modeled as a submodular knapsack set....

Submodular functions describe a variety of discrete problems in machine learn-ing, signal processing, and computer vision. However, minimizing submodularfunctions poses a number of algorithmic challenges. Recent work introduced aneasy-to-use, parallelizable algorithm for minimizing submodular functions thatdecompose as the sum of “simple” submodular functions. Empirically, this ...

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the firs...

Submodular functions occur in many combinatorial optimisation problems and a number of polynomial-time algorithms have been devised to minimise such functions. The time complexity of the fastest known general algorithm for submodular function minimisation (SFM) is O(n6 + n5L), where n is the number of variables and L is the time required to evaluate the function. However, many important special...