{"type":"ScholarlyArticle","title":"Optimizing Ratio of Monotone Set Functions","authors":[{"name":"Chao Qian","slug":"chao-qian","given":"Chao","family":"Qian","affiliation":"UBRI , School of Computer Science and Technology , University of Science and Technology of China , Hefei 230027 , China"},{"name":"Jing-Cheng Shi","slug":"jing-cheng-shi","given":"Jing-Cheng","family":"Shi","affiliation":"National Key Laboratory for Novel Software Technology , Nanjing University , Nanjing 210023 , China"},{"name":"Yang Yu","slug":"yang-yu","given":"Yang","family":"Yu","affiliation":"National Key Laboratory for Novel Software Technology , Nanjing University , Nanjing 210023 , China"},{"name":"Ke Tang","slug":"ke-tang","given":"Ke","family":"Tang","affiliation":"UBRI , School of Computer Science and Technology , University of Science and Technology of China , Hefei 230027 , China"},{"name":"Zhi-Hua Zhou","slug":"zhi-hua-zhou","given":"Zhi-Hua","family":"Zhou","affiliation":"National Key Laboratory for Novel Software Technology , Nanjing University , Nanjing 210023 , China"}],"abstract":"This paper considers the problem of minimizing the ratio of two set functions, i.e., $f/g$. Previous work assumed monotone and submodular of the two functions, while we consider a more general situation where $g$ is not necessarily submodular. We derive that the greedy approach GreedRatio, as a fixed time algorithm, achieves a $\\frac{|X^*|}{(1+(|X^*| \\textendash 1)(1 \\textendash \\kappa_f))\\gamma(g)}$ approximation ratio, which also improves the previous bound for submodular $g$. If more time can be spent, we present the PORM algorithm, an anytime randomized iterative approach minimizing $f$ and $\\textendash g$ simultaneously. We show that PORM using reasonable time has the same general approximation guarantee as GreedRatio, but can achieve better solutions in cases and applications.","doi":"10.24963/ijcai.2017/363","url":"https://ijcai.org/proceedings/2017/363","pdf":"https://ijcai.org/proceedings/2017/0363.pdf","markdown":"https://ijcai.org/proceedings/2017/363.md","bibtex":"https://ijcai.org/proceedings/2017/bibtex/363","volume":{"slug":"2017","title":"Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2017","url":"https://ijcai.org/proceedings/2017/","year":2017,"location":"Melbourne, Australia","dates":"19-25 August 2017","editors":"Carles Sierra","isbn":"978-0-9992411-0-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2017"},"section":"Machine Learning A-R","track":"Main track","number":363,"pages":{"first":2606,"last":2612},"keywords":["Combinatorial & Heuristic Search: Combinatorial search/optimisation","Machine Learning: Machine Learning","Combinatorial & Heuristic Search: Heuristic Search"],"videos":[],"references":[{"position":1,"raw":"[Bai et al., 2016] W. 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