# Reoptimization of the Closest Substring Problem under Pattern Length Modification

## TL;DR

This paper explores reoptimization techniques for the closest substring problem, demonstrating that while the problem remains hard with added pattern length, approximation algorithms can leverage previous solutions to improve efficiency and accuracy.

## Contribution

It introduces greedy approximation algorithms utilizing previous solutions for reoptimization, proving their additive error bounds and improving the PTAS runtime.

## Key findings

- Problem remains hard with k=1.
- Approximation algorithms have additive error increasing with k.
- Reoptimization can slightly improve PTAS runtime.

## Abstract

This study investigates whether reoptimization can help in solving the closest substring problem. We are dealing with the following reoptimization scenario. Suppose, we have an optimal l-length closest substring of a given set of sequences S. How can this information be beneficial in obtaining an (l+k)-length closest substring for S? In this study, we show that the problem is still computationally hard even with k=1. We present greedy approximation algorithms that make use of the given information and prove that it has an additive error that grows as the parameter k increases. Furthermore, we present hard instances for each algorithm to show that the computed approximation ratio is tight. We also show that we can slightly improve the running-time of the existing polynomial-time approximation scheme (PTAS) for the original problem through reoptimization.