On a Speculated Relation Between Chv\'atal-Sankoff Constants of Several Sequences

TL;DR

This paper disproves a conjecture about the relationship between Chvátal-Sankoff constants for multiple sequences and provides new lower bounds for these constants in small alphabet and sequence number cases.

Contribution

It challenges Steele's speculation on the relation between gamma_{2,d} and gamma_{2,2}, and introduces improved lower bounds for gamma_{sigma,d} for small parameters.

Findings

Disproved Steele's conjecture on gamma_{2,d} and gamma_{2,2}.

Derived new lower bounds for gamma_{sigma,d} with small sigma and d.

Clarified the behavior of longest common subsequence constants for small sequence and alphabet sizes.

Abstract

It is well known that, when normalized by n, the expected length of a longest common subsequence of d sequences of length n over an alphabet of size sigma converges to a constant gamma_{sigma,d}. We disprove a speculation by Steele regarding a possible relation between gamma_{2,d} and gamma_{2,2}. In order to do that we also obtain new lower bounds for gamma_{sigma,d}, when both sigma and d are small integers.