The (multivariate) fine-grained complexity of Longest Common Subsequence

Notwithstanding the lack of improvement in the worst case, a long and successful line of research identified and exploited parameters of the input strings, achieving strongly subquadratic running times for highly relevant special cases, e.g., comparing large, but very similar files. The questions arise whether (i) the best known algorithms have an optimal parameter dependence (up to lower order factors, assuming SETH) and whether (ii) some parameter constellations admit faster algorithms than currently known (without refuting SETH). To this end, we provide a \emph{systematic}, effectively tight study of the (conditional) complexity of LCS, taking into account the parameters previously discussed in the literature: n:=max{|x|,|y|}, m:=min{|x|,|y|}, the length L of an LCS of x and y, the alphabet size |\Sigma|, m-L, n-L, as well as the numbers of matching and dominant pairs.

This is joint work with Karl Bringmann.