Arithmetic Expression Construction

TL;DR

This paper investigates the computational complexity of constructing arithmetic expressions from given numbers and operators to reach a target, analyzing various constraints and proving NP-completeness for multiple problem variants.

Contribution

It introduces a unified framework for proving NP-completeness of arithmetic expression construction problems under different constraints.

Findings

Most variants are NP-complete, sometimes in the strong sense.

The rational function framework links these problems to rational functions and positive integers.

Complexity results apply to many subsets of the operators {+, -, ×, ÷}.

Abstract

When can $n$ given numbers be combined using arithmetic operators from a given subset of $\{+, -, \times, \div\}$ 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 $\{+,-,\times,\div\}$, 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.