When Input Integers are Given in the Unary Numeral Representation

TL;DR

This paper investigates how representing input integers in unary form affects the computational complexity of NP-complete problems, revealing that many become easier to solve when given in unary, highlighting structural complexity differences.

Contribution

The work identifies numerous NP-complete problems that are easily solvable with unary input integers and discusses the implications for understanding strong NP-completeness.

Findings

Many NP-complete problems become easier with unary input representation

Unary representation can reduce problems from NP-complete to polynomial-time solvable

The results highlight structural differences between strong and non-strong NP-completeness

Abstract

Many NP-complete problems take integers as part of their input instances. These input integers are generally binarized, that is, provided in the form of the "binary" numeral representation, and the lengths of such binary forms are used as a basis unit to measure the computational complexity of the problems. In sharp contrast, the "unarization" (or the "unary" numeral representation) of numbers has been known to bring a remarkably different effect onto the computational complexity of the problems. When no computational-complexity difference is observed between binarization and unarization of instances, on the contrary, the problems are said to be strong NP-complete. This work attempts to spotlight an issue of how the unarization of instances affects the computational complexity of various combinatorial problems. We present numerous NP-complete (or even NP-hard) problems, which turn out to be easily solvable when input integers are represented in unary. We then discuss the computational complexities of such problems when taking unary-form integer inputs. We hope that a list of such problems signifies the structural differences between strong NP-completeness and non-strong NP-completeness.