An undirected simple graph G is locally irregular if adjacent vertices of G have different
degrees. An edge-colouring ϕ of G is locally irregular if each colour class of ϕ induces a
locally irregular subgraph of G. The irregular chromatic index χ irr′(G) of G is the least
number of colours used by a locally irregular edge-colouring of G (if any). We show that the
problem of determining the irregular chromatic index of a graph can be handled in linear
time when restricted to trees, but it remains NP-complete in general.