Variants of Constrained Longest Common Subsequence

TL;DR

This paper introduces the Doubly-Constrained Longest Common Subsequence (DC-LCS), a new variant of LCS that incorporates multiple constraints relevant to computational biology, and provides algorithms and complexity results.

Contribution

It defines the DC-LCS problem, offers a fixed-parameter algorithm, and establishes its parameterized hardness and NP-hardness under certain conditions.

Findings

Fixed-parameter algorithm for DC-LCS based on solution length

Parameterized hardness result for Constrained LCS with respect to constraints and alphabet size

NP-hardness of DC-LCS when alphabet size is constant

Abstract

In this work, we consider a variant of the classical Longest Common Subsequence problem called Doubly-Constrained Longest Common Subsequence (DC-LCS). Given two strings s1 and s2 over an alphabet A, a set C_s of strings, and a function Co from A to N, the DC-LCS problem consists in finding the longest subsequence s of s1 and s2 such that s is a supersequence of all the strings in Cs and such that the number of occurrences in s of each symbol a in A is upper bounded by Co(a). The DC-LCS problem provides a clear mathematical formulation of a sequence comparison problem in Computational Biology and generalizes two other constrained variants of the LCS problem: the Constrained LCS and the Repetition-Free LCS. We present two results for the DC-LCS problem. First, we illustrate a fixed-parameter algorithm where the parameter is the length of the solution. Secondly, we prove a parameterized hardness result for the Constrained LCS problem when the parameter is the number of the constraint strings and the size of the alphabet A. This hardness result also implies the parameterized hardness of the DC-LCS problem (with the same parameters) and its NP-hardness when the size of the alphabet is constant.