An Efficient Framework for Balancing Submodularity and Cost

Abstract

In the classical selection problem, the input consists of a collection of elements and the goal is to pick a subset of elements from the collection such that some objective function f is maximized. This problem has been studied extensively in the data-mining community and it has multiple applications including influence maximization in social networks, team formation and recommender systems. A particularly popular formulation that captures the needs of many such applications is one where the objective function f is a monotone and non-negative submodular function. In these cases, the corresponding computational problem can be solved using a simple greedy (1 − 1 e )-approximation algorithm. In this paper, we consider a generalization of the above formulation where the goal is to optimize a function that maximizes the submodular function f minus a linear cost function c. This formulation appears as a more natural one, particularly when one needs to strike a balance between the value of the objective function and the cost being paid in order to pick the selected elements. We address variants of this problem both in an offline setting, where the collection is known apriori, as well as in online settings, where the elements of the collection arrive in an online fashion. We demonstrate that by using simple variants of the standard greedy algorithm (used for submodular optimization) we can design algorithms that have provable approximation guarantees, are extremely efficient and work very well in practice. CCS CONCEPTS• Theory of computation → Approximation algorithms analysis.

Maximizing Sums of Non-monotone Submodular and Linear Functions: Understanding the Unconstrained Case

“…In particular, they described algorithms with optimal approximation guarantees for this problem when g is a non-negative monotone submodular function, ℓ is a linear function and the optimization is subject to either a matroid or a cardinality constraint. 1 Later works obtained faster and semi-streaming algorithms for the same setting [7,10,11,14]. However, in contrast to all these (often tight) results for monotone submodular functions g, much less is known about the case of non-monotone submodular functions.…”

On Maximizing Sums of Non-monotone Submodular and Linear Functions

“…Is there an online algorithm that works for general non-monotone and non-positive? We note that the semi-streaming algorithms studied by Kazemi et al [Kaz+21] and Nikolakaki et al [NET21] provide a (0.5, 1)-approximation algorithm for RegularizedUSM when is monotone. For non-monotone USM, simply selecting each element of with probability 0.5 achieves a 0.25-approximation [FMV11].…”

Dynamic Thresholding for Online Distributed Data Selection

“…In particular, [9] demonstrated the possiblity of a greater-than-1/2-approximation in expectation, [10] precisely achieved a (1 − 1/e − )-approximation in O(k2 poly(1/ ) ) memory, and most recently [11] achieved the same constant-factor approximation in improved O(k/ ) memory. Motivated by applications to team formation, [12] introduced a simple and efficient streaming algorithm which approximates an unconstrained, cost-diminished version of (1) for arbitrary D and submodular, nonnegative f in O(k) memory. Though our optimization objective is fundamentally different from theirs (we optimize a submodular function while their objective is the difference between a submodular value function and an additive cost function), our selection rule and theoretical guarantees are inspired by their approach.…”