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Arithmetic Expression Construction

Authors Leo Alcock, Sualeh Asif, Jeffrey Bosboom, Josh Brunner, Charlotte Chen, Erik D. Demaine , Rogers Epstein, Adam Hesterberg, Lior Hirschfeld, William Hu, Jayson Lynch, Sarah Scheffler, Lillian Zhang

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Author Details

Leo Alcock
  • Harvard University, Cambridge, MA, USA
Sualeh Asif
  • MIT, Cambridge, MA, USA
Jeffrey Bosboom
  • CSAIL, MIT, Cambridge, MA, USA
Josh Brunner
  • CSAIL, MIT, Cambridge, MA, USA
Charlotte Chen
  • MIT, Cambridge, MA, USA
Erik D. Demaine
  • CSAIL, MIT, Cambridge, MA, USA
Rogers Epstein
  • CSAIL, MIT, Cambridge, MA, USA
Adam Hesterberg
  • Harvard University, Cambridge, MA, USA
Lior Hirschfeld
  • MIT, Cambridge, MA, USA
William Hu
  • MIT, Cambridge, MA, USA
Jayson Lynch
  • MIT, CSAIL, Cambridge, MA, USA
Sarah Scheffler
  • Boston University, Boston, MA, USA
Lillian Zhang
  • MIT, Cambridge, MA, USA


This work was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.892) taught by Erik Demaine in Spring 2019. We thank the other participants of that class - in particular, Josh Gruenstein, Mirai Ikebuchi, and Vilhelm Andersen Woltz - for related discussions and providing an inspiring atmosphere.

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Leo Alcock, Sualeh Asif, Jeffrey Bosboom, Josh Brunner, Charlotte Chen, Erik D. Demaine, Rogers Epstein, Adam Hesterberg, Lior Hirschfeld, William Hu, Jayson Lynch, Sarah Scheffler, and Lillian Zhang. Arithmetic Expression Construction. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 12:1-12:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


When can n given numbers be combined using arithmetic operators from a given subset of {+,-,×,÷} to obtain a given target number? We study three variations of this problem of Arithmetic Expression Construction: when the expression (1) is unconstrained; (2) has a specified pattern of parentheses and operators (and only the numbers need to be assigned to blanks); or (3) must match a specified ordering of the numbers (but the operators and parenthesization are free). For each of these variants, and many of the subsets of {+,-,×,÷}, we prove the problem NP-complete, sometimes in the weak sense and sometimes in the strong sense. Most of these proofs make use of a rational function framework which proves equivalence of these problems for values in rational functions with values in positive integers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Hardness
  • algebraic complexity
  • expression trees


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