  Longest Run Subsequence is a problem introduced recently in the context of
the scaffolding phase of genome assembly (Schrinner et al., WABI 2020). The
problem asks for a maximum length subsequence of a given string that contains
at most one run for each symbol (a run is a maximum substring of consecutive
identical symbols). The problem has been shown to be NP-hard and to be
fixed-parameter tractable when the parameter is the size of the alphabet on
which the input string is defined. In this paper we further investigate the
complexity of the problem and we show that it is fixed-parameter tractable when
it is parameterized by the number of runs in a solution, a smaller parameter.
Moreover, we investigate the kernelization complexity of Longest Run
Subsequence and we prove that it does not admit a polynomial kernel when
parameterized by the size of the alphabet or by the number of runs. Finally, we
consider the restriction of Longest Run Subsequence when each symbol has at
most two occurrences in the input string and we show that it is APX-hard.
