2020/11/23 by Alcock, Leo, Asif, Sualeh, Bosboom, Jeffrey +10
#Computational Complexity (cs.CC) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2011.11767
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.