Novel algorithms for maximum DS decomposition

• Published in: Theoretical computer science Vol. 857; pp. 87 - 96
• Main Authors: Chen, Shengminjie, Yang, Wenguo, Gao, Suixiang, Jin, Rong
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 12.02.2021
• Subjects: DS decomposition; Non-submodularity; Parameter conditioned greedy; Non-submodularity; Parameter conditioned greedy; DS decomposition
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2020.12.041

•Authors have studied the problem of DS decomposition which is an important set function optimization problem. We have tackled the problem of how to effectively and efficiently solve the problem of maximizing DS decomposition as it is a vital part to solve for any set function.•Deterministic Parameter Conditioned Greedy Algorithm and Random Parameter Conditioned Greedy Algorithm are algorithms we proposed. Also, we have focused on the difference with parameter and combined it with non-negative condition and have obtained some novel approximation as well.•All the proofs, details and algorithms seem to be flexible as the Conditioned Algorithm is the general framework, different users can choose the parameters that fit their problem to get a better approximation. DS decomposition plays an important role in set function optimization problem, because there is DS decomposition for any set function. How to design an efficient and effective algorithm to solve maximizing DS decomposition is a heated problem. In this work, we propose a framework called Parameter Conditioned Greedy Algorithm which has a deterministic version and two random versions. In more detail, this framework uses the difference with parameter decomposition function and combines non-negative condition. Besides, if we set the different parameters, the framework can return solution with different approximation ratio. Also, we choose two special case to show our deterministic algorithm gets f(Sk)−(e−1−cg)g(Sk)≥(1−e−1)[f(OPT)−g(OPT)] and f(Sk)−(1−cg)g(Sk)≥(1−e−1)f(OPT)−g(OPT) respectively for cardinality constrained problem, where cg is the curvature of monotone submodular set function. To speed the deterministic algorithm, we introduce a random sample set whose intersection with the optimal solution is as nonempty as possible. Importantly, it also can get the same approximation ratio as deterministic algorithm under expectation. Further, for maximization DS decomposition without constraint, our another random algorithm gets E[f(Sk)−(e−1−cg)g(Sk)]≥(1−e−1)[f(OPT)−g(OPT)] and E[f(Sk)−(1−cg)g(Sk)]≥(1−e−1)f(OPT)−g(OPT) respectively. Because the Parameter Conditioned Algorithm is the general framework, different users can choose the parameters that fit their problem to get a better approximation.