Our source-code is publicly available at https://github.com/csirac/dynamicDBG under an open-source license.

In this paper, the Minimum Cost Submodular Cover problem is studied, which is to minimize a modular cost function such that the monotone submodular benefit function is above a threshold. For this problem, an evolutionary algorithm EASC is introduced that achieves a constant, bicriteria approximation in expected polynomial time; this is the first polynomial-time evolutionary approximation algorithm for Minimum Cost Submodular Cover. To achieve this running time, ideas motivated by submodularity and monotonicity are incorporated into the evolutionary process, which likely will extend to other submodular optimization problems. In a practical application, EASC is demonstrated to outperform the greedy algorithm and converge faster than competing evolutionary algorithms for this problem. 1 The function c is modular if c(X) = x∈X c({x}) for all X ⊆ S. 2 The greedy algorithm is discussed in Section 5.1 of the Appendix.1

This paper proposes the optimization problem Non-Monotone Submodular Cover (SC), which is to minimize the cost required to ensure that a non-monotone submodular benefit function exceeds a given threshold. Two algorithms are presented for SC that both give a ((1 + )(4/ 2 + 1), 1/2(1 − )) bicriteria approximation guarantee to the problem. Both algorithms process the ground set in a stream, one in multiple passes. Further, a (1/2(1 − ), (1 + )(4/ 2 + 1)) bicriteria approximation guarantee is given for the related optimization problem Submodular Knapsack (SK).

In this paper, the monotone submodular maximization problem (SM) is studied. SM is to find a subset of size kappa from a universe of size n that maximizes a monotone submodular objective function f . We show using a novel analysis that the Pareto optimization algorithm achieves a worst-case ratio of (1 − epsilon)(1 − 1/e) in expectation for every cardinality constraint kappa < P , where P ≤ n + 1 is an input, in O(nP ln(1/epsilon)) queries of f . In addition, a novel evolutionary algorithm called the biased Pareto optimization algorithm, is proposed that achieves a worst-case ratio of (1 − epsilon)(1 − 1/e − epsilon) in expectation for every cardinality constraint kappa < P in O(n ln(P ) ln(1/epsilon)) queries of f . Further, the biased Pareto optimization algorithm can be modified in order to achieve a a worst-case ratio of (1 − epsilon)(1 − 1/e − epsilon) in expectation for cardinality constraint kappa in O(n ln(1/epsilon)) queries of f . An empirical evaluation corroborates our theoretical analysis of the algorithms, as the algorithms exceed the stochastic greedy solution value at roughly when one would expect based upon our analysis.

The optimization of submodular functions on the integer lattice has received much attention recently, but the objective functions of many applications are nonsubmodular. We provide two approximation algorithms for maximizing a nonsubmodular function on the integer lattice subject to a cardinality constraint; these are the first algorithms for this purpose that have polynomial query complexity. We propose a general framework for influence maximization on the integer lattice that generalizes prior works on this topic, and we demonstrate the efficiency of our algorithms in this context.