# Linearly-growing Reductions of Karp's 21 NP-complete Problems

## TL;DR

This paper demonstrates that 21 classical NP-complete problems can be reduced to a core set of six problems with only linear growth in size, facilitating more efficient problem solving and potential applications in optimization.

## Contribution

It identifies a kernel subset of six NP-complete problems to which the entire set of 21 can be linearly reduced, a novel insight into problem reduction.

## Key findings

- Six problems form a kernel subset with linear reduction from 21 problems
- Reductions include 0-1 integer programming, job sequencing, Hamiltonian cycle
- Potential for improved optimization problem solving

## Abstract

We address the question of whether it may be worthwhile to convert certain, now classical, NP-complete problems to one of a smaller number of kernel NP-complete problems. In particular, we show that Karp's classical set of 21 NP-complete problems contains a kernel subset of six problems with the property that each problem in the larger set can be converted to one of these six problems with only linear growth in problem size. This finding has potential applications in optimisation theory because the kernel subset includes 0-1 integer programming, job sequencing and undirected Hamiltonian cycle problems.