# A Note on the Inapproximability of Induced Disjoint Paths

## TL;DR

This paper proves that approximating the induced disjoint paths problem within a factor of n^{1-ε} is NP-hard, extending known hardness results and providing a simple reduction from the independent set problem.

## Contribution

It establishes a new inapproximability bound for the induced disjoint paths problem using a straightforward reduction from the independent set problem.

## Key findings

- NP-hard to approximate within n^{1-ε} for any ε>0
- Reduces from the independent set problem to show hardness
- Extends previous hardness results for the problem

## Abstract

We study the inapproximability of the induced disjoint paths problem on an arbitrary $n$-node $m$-edge undirected graph, which is to connect the maximum number of the $k$ source-sink pairs given in the graph via induced disjoint paths. It is known that the problem is NP-hard to approximate within $m^{{1\over 2}-\varepsilon}$ for a general $k$ and any $\varepsilon>0$. In this paper, we prove that the problem is NP-hard to approximate within $n^{1-\varepsilon}$ for a general $k$ and any $\varepsilon>0$ by giving a simple reduction from the independent set problem.