An Efficient Dynamic Programming Algorithm for the Generalized LCS Problem with Multiple Substring Exclusion Constrains

TL;DR

This paper introduces a new dynamic programming algorithm for the generalized longest common subsequence problem with multiple substring exclusion constraints, demonstrating it is not NP-hard and providing an efficient solution.

Contribution

The paper presents a novel dynamic programming algorithm that efficiently solves the generalized LCS problem with multiple substring exclusion constraints, challenging previous NP-hard assumptions.

Findings

The new algorithm correctly computes the LCS with constraints.

Time complexity of the algorithm is O(nmr).

The problem is not NP-hard as previously claimed.

Abstract

In this paper, we consider a generalized longest common subsequence problem with multiple substring exclusion constrains. For the two input sequences $X$ and $Y$ of lengths $n$ and $m$, and a set of $d$ constrains $P=\{P_1,...,P_d\}$ of total length $r$, the problem is to find a common subsequence $Z$ of $X$ and $Y$ excluding each of constrain string in $P$ as a substring and the length of $Z$ is maximized. The problem was declared to be NP-hard\cite{1}, but we finally found that this is not true. A new dynamic programming solution for this problem is presented in this paper. The correctness of the new algorithm is proved. The time complexity of our algorithm is $O(nmr)$.