On the String Consensus Problem and the Manhattan Sequence Consensus Problem

TL;DR

This paper presents a fast, practical algorithm for solving the Manhattan Sequence Consensus problem for up to five sequences, improving efficiency over previous methods and establishing fixed-parameter tractability.

Contribution

The paper introduces an $O(l)$-time algorithm for MSC with up to five sequences and demonstrates fixed-parameter tractability for general $k$ via linear kernelization.

Findings

Efficient $O(l)$-time algorithm for $k \\le 5$ sequences.

Experimental validation of practicality.

Reduction to a kernel of size $k!$ for fixed-parameter tractability.

Abstract

In the Manhattan Sequence Consensus problem (MSC problem) we are given $k$ integer sequences, each of length $l$, and we are to find an integer sequence $x$ of length $l$ (called a consensus sequence), such that the maximum Manhattan distance of $x$ from each of the input sequences is minimized. For binary sequences Manhattan distance coincides with Hamming distance, hence in this case the string consensus problem (also called string center problem or closest string problem) is a special case of MSC. Our main result is a practically efficient $O(l)$-time algorithm solving MSC for $k\le 5$ sequences. Practicality of our algorithms has been verified experimentally. It improves upon the quadratic algorithm by Amir et al.\ (SPIRE 2012) for string consensus problem for $k=5$ binary strings. Similarly as in Amir's algorithm we use a column-based framework. We replace the implied general integer linear programming by its easy special cases, due to combinatorial properties of the MSC for $k\le 5$. We also show that for a general parameter $k$ any instance can be reduced in linear time to a kernel of size $k!$, so the problem is fixed-parameter tractable. Nevertheless, for $k\ge 4$ this is still too large for any naive solution to be feasible in practice.