This question concerns right angled Artin groups (RAAGs), also called partially commutative groups or graph groups.

A student of mine, Adi Ben-Zvi, needs for an algorithm in RAAGs, a subalgorithm that we suspect should be known.

In RAAGs, it is known how to compute efficiently the centralizer of a single element [A Baudisch, Subgroups of semifree groups, Acta Math. Acad. Sci. Hungar. 38 (1981) 19–28; H Servatius, Automorphisms of Graph Groups, J. Algebra 126 (1987) 34–60]. In these papers, the cyclically reduced case is treated, but this obviously implies - since we consider a single element - the case of a general element.

Question: Is there an efficient algorithm for computing, in RAAGs, the centralizer of a given finite set of (general) group elements?

We have a positive answer for sets of cyclically reduced elements, but thus far we only have special cases of the general case.

If no such algorithm is known, we would appreciate even results providing information about centralizers of given finite sets.

Update: Following @YCor, by "computing" we mean providing a set of generators (elements of the original RAAG).