On Subset Selection with General Cost Constraints

Abstract

This paper considers the subset selection problem with a monotone objective function and a monotone cost constraint, which relaxes the submodular property of previous studies. We first show that the approximation ratio of the generalized greedy algorithm is α 2 (1 − 1 e α ) (where α is the submodularity ratio); and then propose POMC, an anytime randomized iterative approach that can utilize more time to find better solutions than the generalized greedy algorithm. We show that POMC can obtain the same general approximation guarantee as the generalized greedy algorithm, but can achieve better solutions in cases and applications.

Pareto Optimization for Subset Selection with Dynamic Cost Constraints

“…It has been shown in [20] that the generalized greedy algorithm, which adds the item with the highest marginal contribution to the current solution in each step, achieves a (1/2)(1 − 1 e )-approximate solution if f is monotone and submodular. [15] extended these results to objective functions with α f submodularity ratio and proved that the generalized greedy algorithm obtains a φ = (α f /2)(1 − 1 e α f )-approximation. For the remainder of this paper, we assume φ = (α f /2)(1 − 1 e α f ) and are interested in obtaining solutions that are φ-approximation for the considered problems.…”

“…In this section we analyze the behavior of POMC facing a dynamic change. According to Lemma 3 in [15], we have for any X ⊆ V and v * = arg max v /…”

“…We compare the generalized greedy algorithm (GGA) and adaptive generalized greedy algorithm (AGGA) with the POMC algorithm on the submodular influence maximization problem [20,15]. We consider dynamic variants of the problems where the constraint bound changes over time.…”

“…For more detailed descriptions of the influence maximization through a social network problem we refer the reader to [9,20,15].…”

An Efficient Evolutionary Algorithm for Minimum Cost Submodular Cover

“…Hence in POM, the population can get quite large, which in turn affects the expected time before the approximation ratio is reached. A similar issue arises when an EA for the dual problem of MCSC is analyzed [Qian et al, 2017]. EASC does not have this problem as its population size is always O((c max /c min ) ln(1/ )n).…”