Submodular functions are discrete analogue of convex functions, arising in various fields of applied mathematics including game theory, information theory, and queueing theory. This survey aims at providing an overview on fundamental properties of submodular functions and recent algorithmic developments of their optimization and approximation. For submodular function minimization, the ellipsoid...

Many combinatorial optimization problems have underlying goal functions that are submodular. The classical goal is to find a good solution for a given submodular function f under a given set of constraints. In this paper, we investigate the runtime of a multi-objective evolutionary algorithm called GSEMO until it has obtained a good approximation for submodular functions. For the case of monoto...

The ability to integrate information in the brain is considered to be an essential property for cognition and consciousness. Integrated Information Theory (IIT) hypothesizes that the amount of integrated information (Φ) in the brain is related to the level of consciousness. IIT proposes that to quantify information integration in a system as a whole, integrated information should bemeasured acr...

We investigate two novel mixed robust/average-case submodular data partitioning problems that we collectively call Submodular Partitioning. These problems generalize purely robust instances of the problem, namely max-min submodular fair allocation (SFA) [8] and min-max submodular load balancing (SLB) [15], and also average-case instances, that is the submodular welfare problem (SWP) [16] and su...

Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization. In this paper, we consider the problem of approxi...

We consider the closely related problems of maximizing a submodular function subject to a cardinality constraint and that of maximizing the sum of submodular functions subject to a partition matroid contraint. Motivated by applications in team selection, we focus on settings where the function(s) can be expressed as the expected value of a symmetric submodular value function of a set of indepen...

Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algor...

We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0, 1}. Our main result is the following structural theorem: any submodular function is -close in `2 to a real-valued decision tree (DT) of depth O(1/ ). This immediately implies that any submodular function is -close to a function of at most 2 ) vari...

This thesis is a detailed examination of the expressive power of valued constraints and related complexity questions. The valued constraint satisfaction problem (VCSP) is a generalisation of the constraint satisfaction problem which allows a variety of combinatorial optimisation problems to be described. Although most results are stated in this framework, they can be interpreted equivalently in...

In this technical report, we aim to give a simple yet detailed analysis of several various submodular maximization algorithms. We start from analyzing the classical greedy algorithm, firstly discussed by Nemhauser et al. (1978), that guarantees a tight bound for constrained maximization of monotonically submodular set functions. We then continue by discussing two randomized algorithms proposed ...