eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
12:1
12:15
10.4230/LIPIcs.ISAAC.2020.12
article
Arithmetic Expression Construction
Alcock, Leo
1
Asif, Sualeh
2
Bosboom, Jeffrey
3
Brunner, Josh
3
Chen, Charlotte
2
Demaine, Erik D.
3
https://orcid.org/0000-0003-3803-5703
Epstein, Rogers
3
Hesterberg, Adam
1
Hirschfeld, Lior
2
Hu, William
2
Lynch, Jayson
4
Scheffler, Sarah
5
Zhang, Lillian
2
Harvard University, Cambridge, MA, USA
MIT, Cambridge, MA, USA
CSAIL, MIT, Cambridge, MA, USA
MIT, CSAIL, Cambridge, MA, USA
Boston University, Boston, MA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol181-isaac2020/LIPIcs.ISAAC.2020.12/LIPIcs.ISAAC.2020.12.pdf
Hardness
algebraic complexity
expression trees