Arithmetic expression construction

• Main Authors: Alcock, L (Author), Bosboom, J (Author), Chen, C (Author), Epstein, R (Author), Hirschfeld, L (Author), Lynch, J (Author), Zhang, L (Author), Asif, S (Author), Brunner, J (Author), Demaine, ED (Author), Hesterberg, A (Author), Hu, W (Author), Scheffler, S (Author)
• Format: Article
• Language: English
• Published: 2022-07-22T14:19:40Z.
• Subjects: Article
• Online Access: Get fulltext

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.