Monotone Nonsubmodular Maximization

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Presentation transcript:

Monotone Nonsubmodular Maximization Supermodular Degree

k-Extendible System

Independent System Consider a finite set V and a collection of subsets of V. (V, ) is called an independent system if i.e., it is hereditary. Each subset in is called an independent set.

Matroid

k-extendible

k-intersection

Lemma 1

Supermodular Degree M. Feldman and R. Izsak, “Constrained monotone function maximization and the supermodular degree,” in ACM-SIAM SODA, 2014.

Definition

Monotone Function Max

Greedy Approximation Theorem 1

Proof

Monotone Function Max

Greedy Approximation Theorem 1

Proof Monotone!

Applications Uriel Feige and Rani Izsak. Welfare maximization and the supermodular degree. In ITCS, pages 247–256, 2013. Rani Izsak: Working Together: Committee Selection and the Supermodular Degree. AAMAS 2017: 1578-1580 Moran Feldman, Rani Izsak: Building a Good Team: Secretary Problems and the Supermodular Degree. SODA 2017: 1651-1670

Submodularity Ratio and Curvature Andrew An Bian, Joachim M. Buhmann, Andreas Krause,Sebastian Tschiatschek: Guarantees for Greedy Maximization of Non-submodular Functions with Applications, arXiv:1703.02100v3 [cs.DM] 13 June 2017

Submodularity Ratio

Curvature

Greedy Theorem

Thank You, end