# A $(1.4 + \epsilon)$-approximation algorithm for the $2$-Max-Duo problem

## TL;DR

This paper introduces a new approximation algorithm for the 2-Max-Duo problem, achieving a ratio close to 1.4, improving upon previous results and leveraging a vertex-degree reduction technique.

## Contribution

The paper presents a novel vertex-degree reduction method that enables a $(1.4 + ext{epsilon})$-approximation for the 2-Max-Duo problem, surpassing prior approximation ratios.

## Key findings

- Achieved a $(1.4 + ext{epsilon})$-approximation for 2-Max-Duo.
- Developed a vertex-degree reduction technique.
- Improved approximation ratio from previous 1.6+epsilon.

## Abstract

The maximum duo-preservation string mapping (Max-Duo) problem is the complement of the well studied minimum common string partition (MCSP) problem, both of which have applications in many fields including text compression and bioinformatics. $k$-Max-Duo is the restricted version of Max-Duo, where every letter of the alphabet occurs at most $k$ times in each of the strings, which is readily reduced into the well known maximum independent set (MIS) problem on a graph of maximum degree $\Delta \le 6(k-1)$. In particular, $2$-Max-Duo can then be approximated arbitrarily close to $1.8$ using the state-of-the-art approximation algorithm for the MIS problem. $2$-Max-Duo was proved APX-hard and very recently a $(1.6 + \epsilon)$-approximation was claimed, for any $\epsilon > 0$. In this paper, we present a vertex-degree reduction technique, based on which, we show that $2$-Max-Duo can be approximated arbitrarily close to $1.4$.