Faster Guarantees of Evolutionary Algorithms for Maximization of Monotone Submodular Functions

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.

An optimization problem for continuous submodular functions

"Real continuous submodular functions, as a generalization of the corresponding discrete notion to the continuous domain, gained considerable attention recently. The analog notion for entropy functions requires additional properties: a real function defined on the non-negative orthant of $\R^n$ is entropy-like (EL) if it is submodular, takes zero at zero, non-decreasing, and has the Diminishing Returns property. Motivated by problems concerning the Shannon complexity of multipartite secret sharing, a special case of the following general optimization problem is considered: find the minimal cost of those EL functions which satisfy certain constraints. In our special case the cost of an EL function is the maximal value of the $n$ partial derivatives at zero. Another possibility could be the supremum of the function range. The constraints are specified by a smooth bounded surface $S$ cutting off a downward closed subset. An EL function is feasible if at the internal points of $S$ the left and right partial derivatives of the function differ by at least one. A general lower bound for the minimal cost is given in terms of the normals of the surface $S$. The bound is tight when $S$ is linear. In the two-dimensional case the same bound is tight for convex or concave $S$. It is shown that the optimal EL function is not necessarily unique. The paper concludes with several open problems."

Structured Robust Submodular Maximization: Offline and Online Algorithms

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. Although these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids and knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem are known in advance and the online setting where the input data are revealed over time. For the offline setting, we give a general (nearly) optimal bicriteria approximation algorithm that relies on new extensions of classical algorithms for submodular maximization. For the online version of the problem, we give an algorithm that returns a bicriteria solution with sublinear regret. Summary of Contribution: Constrained submodular maximization is one of the core areas in combinatorial optimization with a wide variety of applications in operations research and computer science. Over the last decades, both communities have been interested on the design and analysis of new algorithms with provable guarantees. Sensor location, influence maximization and data summarization are some of the applications of submodular optimization that lie at the intersection of the aforementioned communities. Particularly, our work focuses on optimizing several submodular functions simultaneously. We provide new insights and algorithms to the offline and online variants of the problem which significantly expand the related literature. At the same time, we provide a computational study that supports our theoretical results.

Maximizing Sequence-Submodular Functions and Its Application to Online Advertising

Motivated by applications in online advertising, we consider a class of maximization problems where the objective is a function of the sequence of actions and the running duration of each action. For these problems, we introduce the concepts of sequence-submodularity and sequence-monotonicity, which extend the notions of submodularity and monotonicity from functions defined over sets to functions defined over sequences. We establish that if the objective function is sequence-submodular and sequence-nondecreasing, then there exists a greedy algorithm that achieves [Formula: see text] of the optimal solution. We apply our algorithm and analysis to two applications in online advertising: online ad allocation and query rewriting. We first show that both problems can be formulated as maximizing nondecreasing sequence-submodular functions. We then apply our framework to these two problems, leading to simple greedy approaches with guaranteed performances. In particular, for the online ad allocation problem, the performance of our algorithm is [Formula: see text], which matches the best known existing performance, and for the query rewriting problem, the performance of our algorithm is [Formula: see text], which improves on the best known existing performance in the literature. This paper was accepted by Chung Piaw Teo, optimization.

Non-submodular model for group profit maximization problem in social networks

In social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions are made by simply majority, which means that most of the users in this group reach an agreement, such as US Presidential Elections. A group is called activated if $$\beta$$ β percent of users are influenced in the group. Enterprise will gain income from all influenced groups. Simultaneously, to propagate influence, enterprise needs pay advertisement diffusion cost. Group profit maximization (GPM) problem aims to pick k seeds to maximize the expected profit that considers the benefit of influenced groups with the diffusion cost. GPM is proved to be NP-hard and the objective function is proved to be neither submodular nor supermodular. An upper bound and a lower bound which are difference of two submodular functions are designed. We propose a submodular–modular algorithm (SMA) to solve the difference of two submodular functions and SMA is shown to converge to a local optimal. We present an randomized algorithm based on weighted group coverage maximization for GPM and apply sandwich framework to get theoretical results. Our experiments verify the efficiency of our methods.