Generalized assignment via submodular optimization with reserved capacity

Ariel Kulik, Kanthi Sarpatwar, Baruch Schieber, Hadas Shachnai

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We study a variant of the generalized assignment problem (GAP) with group constraints. An instance of Group GAP is a set I of items, partitioned into L groups, and a set of m uniform (unit-sized) bins. Each item i ϵ I has a size si > 0, and a profit pi,j ≥ 0 if packed in bin j. A group of items is satisfied if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total profit from satisfied groups is maximized. We point to central applications of Group GAP in Video-on-Demand services, mobile Device-to-Device network caching and base station cooperation in 5G networks. Our main result is a 1/6-approximation algorithm for Group GAP instances where the total size of each group is at most m/n. At the heart of our algorithm lies an interesting derivation of a submodular function from the classic LP formulation of GAP, which facilitates the construction of a high profit solution utilizing at most half the total bin capacity, while the other half is reserved for later use. In particular, we give an algorithm for submodular maximization subject to a knapsack constraint, which finds a solution of profit at least 1/3 of the optimum, using at most half the knapsack capacity, under mild restrictions on element sizes. Our novel approach of submodular optimization subject to a knapsack with reserved capacity constraint may find applications in solving other group assignment problems.

Original languageEnglish (US)
Title of host publication27th Annual European Symposium on Algorithms, ESA 2019
EditorsMichael A. Bender, Ola Svensson, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771245
DOIs
StatePublished - Sep 2019
Event27th Annual European Symposium on Algorithms, ESA 2019 - Munich/Garching, Germany
Duration: Sep 9 2019Sep 11 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume144
ISSN (Print)1868-8969

Conference

Conference27th Annual European Symposium on Algorithms, ESA 2019
CountryGermany
CityMunich/Garching
Period9/9/199/11/19

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Approximation algorithms
  • Group Generalized assignment problem
  • Knapsack constraints
  • Submodular maximization

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