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Scheduling Lower Bounds via AND Subset Sum

Authors Amir Abboud, Karl Bringmann, Danny Hermelin, Dvir Shabtay



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Amir Abboud
  • IBM Almaden Research Center, San Jose, CA, USA
Karl Bringmann
  • Saarland University, Saarland Informatics Campus (SIC), Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrücken, Germany
Danny Hermelin
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beersheba, Israel
Dvir Shabtay
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beersheba, Israel

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Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. Scheduling Lower Bounds via AND Subset Sum. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 4:1-4:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.4

Abstract

Given N instances (X_1,t_1),…,(X_N,t_N) of Subset Sum, the AND Subset Sum problem asks to determine whether all of these instances are yes-instances; that is, whether each set of integers X_i has a subset that sums up to the target integer t_i. We prove that this problem cannot be solved in time Õ((N ⋅ t_max)^{1-ε}), for t_max = max_i t_i and any ε > 0, assuming the ∀ ∃ Strong Exponential Time Hypothesis (∀∃-SETH). We then use this result to exclude Õ(n+P_max⋅n^{1-ε})-time algorithms for several scheduling problems on n jobs with maximum processing time P_max, assuming ∀∃-SETH. These include classical problems such as 1||∑ w_jU_j, the problem of minimizing the total weight of tardy jobs on a single machine, and P₂||∑ U_j, the problem of minimizing the number of tardy jobs on two identical parallel machines.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • SETH
  • fine grained complexity
  • Subset Sum
  • scheduling

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