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Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping

Authors Yaron Fairstein, Ariel Kulik, Hadas Shachnai

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Yaron Fairstein
  • Computer Science Department, Technion, Haifa, Israel
Ariel Kulik
  • Computer Science Department, Technion, Haifa, Israel
Hadas Shachnai
  • Computer Science Department, Technion, Haifa, Israel

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Yaron Fairstein, Ariel Kulik, and Hadas Shachnai. Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ESA.2021.41

Abstract

A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins which adheres to bin capacities. In this paper we present a unified algorithm that yields efficient approximations for a wide class of submodular and modular optimization problems involving multiple knapsack constraints. One notable example is a polynomial time approximation scheme for Multiple-Choice Multiple Knapsack, improving upon the best known ratio of 2. Another example is Non-monotone Submodular Multiple Knapsack, for which we obtain a (0.385-ε)-approximation, matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest.

Subject Classification

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

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