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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 maximumManhattan distance of $x$ from each of the input sequences is minimized. Forbinary sequences Manhattan distance coincides with Hamming distance, hence inthis case the string consensus problem (also called string center problem orclosest string problem) is a special case of MSC. Our main result is apractically efficient $O(l)$-time algorithm solving MSC for $k\le 5$ sequences.Practicality of our algorithms has been verified experimentally. It improvesupon the quadratic algorithm by Amir et al.\ (SPIRE 2012) for string consensusproblem for $k=5$ binary strings. Similarly as in Amir's algorithm we use acolumn-based framework. We replace the implied general integer linearprogramming by its easy special cases, due to combinatorial properties of theMSC for $k\le 5$. We also show that for a general parameter $k$ any instancecan be reduced in linear time to a kernel of size $k!$, so the problem isfixed-parameter tractable. Nevertheless, for $k\ge 4$ this is still too largefor any naive solution to be feasible in practice.

On the String Consensus Problem and the Manhattan Sequence Consensus Problem