Approximation Algorithms For Submodular Optimization And Applications

Abstract

This study proposes approximation algorithms by using several strategies such as streaming, improved-greedy, stop-and-stare, and reverse influence sampling ($ \RIS $) to solve three variants of the submodular optimization problem, and perform experiments of these algorithms on the well-known application problems of submodular optimization such as Influence Threshold ($ \IT $) and Influence Maximization ($ \IM) $. Specifically, in the first problem, we propose the two single-pass streaming algorithms ($ \StrA $ and $ \StrM $) for minimizing the cost of the submodular cover problem under the multiplicative and additive noise models. $ \StrA $ and $ \StrM $ provide bicriteria approximation solutions. These algorithms effectively increase performance computing the objective function, reduce complexity, and apply to big data. For the second problem, we focus on maximizing a submodular function on fairness constraints. This problem is also known as the problem of fairness budget distribution for influence maximization. We design three algorithms ($ \FBIM1 $, $ \FBIM2 $, and $ \FBIM3 $) by combining several strategies such as the threshold greedy algorithm, dynamic stop-and-stare technique, generating samplings by reverse influence sampling framework, and seeds selection to ensure max coverage. $ \FBIM1 $, $ \FBIM2 $, and $ \FBIM3 $ perform effectively on big data, provide $(1/2-\epsilon)$-approximation to the optimum solutions, and require complexities of the comparison algorithms. Finally, we devise two effective streaming algorithm ($ \StrI $ and $ \StrII $) to maximize the Diminishing Returns submodular (DR-submodular) function with a cardinality constraint on the integer lattice for the third problem. $ \StrI $ and $ \StrII $ provide $ (1/2-\epsilon)$-approximation ratio and $ (1-1/e-\epsilon)$-approximation ratio, respectively. Simultaneously, compared with the state-of-the-art, these two algorithms have reduced complexity, superior runtime performance, and negligible difference in objective function values. In each problem, we further investigate the performance of our proposed algorithms by conducting many experiments. The experimental results have indicated that our approximation algorithms provide high-efficiency solutions, outperform the-state-of-art algorithms in complexity, runtime, and satisfy the specified constraints. Some of the results have been confirmed through five publications at the Scopus international conferences (RIVF 2021, ICABDE 2021) and the SCIE journals (Computer Standards $ \& $ Interfaces (Elsevier) and Mathematics (MDPI)).