Better Heuristic Algorithms for the Repetition Free LCS and Other Variants

Abstract

In Discrete Applied Mathematics 2010, Adi et al. introduce and study a variant of the well known Longest Common Subsequence problem, named Repetition Free Longest Common Subsequence (RFLCS). In RFLCS the input consists of two strings A and B over an alphabet $\varSigma$ and the goal is to find the longest common subsequence containing only distinct characters from $\varSigma$. Adi et al. prove that the problem is $\mathcal {APX}$-hard and show three approximation algorithms. Castelli et al. (Operations Research Letters 2013) propose a heuristic genetic algorithm and Blum and Blesa introduce metaheuristic algorithms (International Conference on Artificial Evolution 2013 and Evolutionary Computation in Combinatorial Optimization 2016).

In this paper we design and test several new heuristic algorithms for RFLCS. The first algorithm, uses dynamic programming and in our testing setup outperforms the algorithms of Adi et al. The second heuristic algorithm improves upon the first and becomes comparable to the state-of-the-art algorithms of Blum and Blesa. The third algorithm transforms the RFLCS instance into an instance of the Maximum Independent Set (MIS) problem with the same value of the optimum solution. Then, we apply known algorithms for the MIS problem. We also augment one of the approximation algorithms of Adi et al. and we prove that we achieve an approximation of factor $2\sqrt{\min \{|A|,|B|\}}$.

Finally, we introduce a new variant of the LCS problem, named Multiset Restricted Common Subsequence (MRCS), that is a generalization of RFLCS. We present an exact polynomial time algorithm for MRCS for constant size alphabet. Additionally, we show that MRCS admits a $2\sqrt{\min \{|A|,|B|\}}$ approximation.

This work was supported by the research programme PN 1819 “Advanced IT resources to support digital transformation processes in the economy and society - RESINFO-TD” (2018), project PN 1819-01-01 “New research in complex systems modelling and optimization with applications in industry, business and cloud computing”, funded by the Ministry of Research and Innovation.