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# Correlated Stochastic Knapsack with a Submodular Objective

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## Acknowledgements

We would like to thank the reviewers for comments on a previous version of this draft, who found a flaw in the algorithm and its analysis. The old algorithm imposes the partition matroid while doing the first rounding step, leading to dependency between items. This seemingly convenient step actually breaks the correctness of contention resolution scheme, which is built on FKG inequality and intrinsically needs an independent rounding step. We fixed the issue by replacing it with a true independent rounding step, and fix the solution to fit the partition matroid later on. While this breaks the symmetry between items, the gap of 2 turns out to be large enough to fix everything. Check the use of union bound in Case 1 for the proof of Lemma 10 for details.

## Cite As

Sheng Yang, Samir Khuller, Sunav Choudhary, Subrata Mitra, and Kanak Mahadik. Correlated Stochastic Knapsack with a Submodular Objective. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 91:1-91:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.91

## Abstract

We study the correlated stochastic knapsack problem of a submodular target function, with optional additional constraints. We utilize the multilinear extension of submodular function, and bundle it with an adaptation of the relaxed linear constraints from Ma [Mathematics of Operations Research, Volume 43(3), 2018] on correlated stochastic knapsack problem. The relaxation is then solved by the stochastic continuous greedy algorithm, and rounded by a novel method to fit the contention resolution scheme (Feldman et al. [FOCS 2011]). We obtain a pseudo-polynomial time (1 - 1/√e)/2 ≃ 0.1967 approximation algorithm with or without those additional constraints, eliminating the need of a key assumption and improving on the (1 - 1/∜e)/2 ≃ 0.1106 approximation by Fukunaga et al. [AAAI 2019].

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Rounding techniques
• Theory of computation → Stochastic approximation
• Theory of computation → Stochastic control and optimization
• Theory of computation → Submodular optimization and polymatroids
##### Keywords
• Stochastic Knapsack
• Submodular Optimization
• Stochastic Optimization

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