Is this a known combinatorial optimization/scheduling problem?
Your problem is equivalent to Shortest Common Supersequence (SCS) and was considered in $[1]$ by the name Scheduling on batch machines with precedence constraints as chains and compatibilities. If items in the problem are of the same color then such problem is in P and can be solved in $O(n^2)$  $[2]$.

What concerns approximability a good source is A compendium of NP optimization problems.

The lastest results on SCS can be found in $[3,4]$.

For practical algorithms see $[5]$ whose author states that "Hybrid MA-BS is a current state-of-the-art technique for the SCSP $[6]$ ".


Brauner N., Naves G. Scheduling chains of operations on a batching machine with disjoint sets of operation compatibility.
Brucker P. Scheduling algorithms (Chapter 8. Batching Problems).
Timkovsky V. G. Some Approximations for Shortest Common Nonsubsequences and Supersequences.
Gotthilf Z. and Lewenstein M. Improved Approximation Results on the Shortest Common Supersequence Problem.
Kubalik J. Efficient stochastic local search algorithm for solving the shortest common supersequence problem.
Blum, C., Cotta, C., Fernandez, A. J., Gallardo, J. E. A Probabilistic Beam Search Approach to the Shortest Common Supersequence Problem.