# Heuristic algorithms for the Longest Filled Common Subsequence Problem

## TL;DR

This paper explores heuristic algorithms for the NP-hard Longest Filled Common Subsequence Problem, comparing their performance with existing approximation methods and using ILP for exact solutions on smaller instances.

## Contribution

It introduces new heuristic algorithms, implements and tests them, and develops an ILP model to find exact solutions for moderate-sized instances.

## Key findings

- Heuristic algorithms perform competitively with approximation algorithms.
- ILP approach yields exact solutions for small to medium instances.
- Experimental results demonstrate the effectiveness of the proposed heuristics.

## Abstract

At CPM 2017, Castelli et al. define and study a new variant of the Longest Common Subsequence Problem, termed the Longest Filled Common Subsequence Problem (LFCS). For the LFCS problem, the input consists of two strings $A$ and $B$ and a multiset of characters $\mathcal{M}$. The goal is to insert the characters from $\mathcal{M}$ into the string $B$, thus obtaining a new string $B^*$, such that the Longest Common Subsequence (LCS) between $A$ and $B^*$ is maximized. Casteli et al. show that the problem is NP-hard and provide a 3/5-approximation algorithm for the problem. In this paper we study the problem from the experimental point of view. We introduce, implement and test new heuristic algorithms and compare them with the approximation algorithm of Casteli et al. Moreover, we introduce an Integer Linear Program (ILP) model for the problem and we use the state of the art ILP solver, Gurobi, to obtain exact solution for moderate sized instances.