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A (1-e^{-1}-ε)-Approximation for the Monotone Submodular Multiple Knapsack Problem

Authors Yaron Fairstein, Ariel Kulik, Joseph (Seffi) Naor, Danny Raz, Hadas Shachnai

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Author Details

Yaron Fairstein
  • Computer Science Department, Technion, Haifa, Israel
Ariel Kulik
  • Computer Science Department, Technion, Haifa, Israel
Joseph (Seffi) Naor
  • Computer Science Department, Technion, Haifa, Israel
Danny Raz
  • Computer Science Department, Technion, Haifa, Israel
Hadas Shachnai
  • Computer Science Department, Technion, Haifa, Israel

Funding

  • Naor, Joseph (Seffi): {This research was supported in part by US-Israel BSF grant 2018352 and by ISF grant 2233/19 (2027511).}

Cite AsGet BibTex

Yaron Fairstein, Ariel Kulik, Joseph (Seffi) Naor, Danny Raz, and Hadas Shachnai. A (1-e^{-1}-ε)-Approximation for the Monotone Submodular Multiple Knapsack Problem. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.44

Abstract

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint (SMKP). The input is a set I of items, each associated with a non-negative weight, and a set of bins having arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a subset of items A ⊆ I and a packing of these items in the bins, such that f(A) is maximized. SMKP is a natural extension of both Multiple Knapsack and the problem of monotone submodular maximization subject to a knapsack constraint. Our main result is a nearly optimal polynomial time (1-e^{-1}-ε)-approximation algorithm for the problem, for any ε > 0. Our algorithm relies on a refined analysis of techniques for constrained submodular optimization combined with sophisticated application of tools used in the development of approximation schemes for packing problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
Keywords
  • Sumodular Optimization
  • Multiple Knapsack
  • Randomized Rounding

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