  Techniques based on $k$-th order Hodge Laplacian operators $L_k$ are widely
used to describe the topology as well as the governing dynamics of high-order
systems modeled as simplicial complexes. In all of them, it is required to
solve a number of least square problems with $L_k$ as coefficient matrix, for
example in order to compute some portions of the spectrum or integrate the
dynamical system. In this work, we introduce the notion of optimal collapsible
subcomplex and we present a fast combinatorial algorithm for the computation of
a sparse Cholesky-like preconditioner for $L_k$ that exploits the topological
structure of the simplicial complex. The performance of the preconditioner is
tested for conjugate gradient method for least square problems (CGLS) on a
variety of simplicial complexes with different dimensions and edge densities.
We show that, for sparse simplicial complexes, the new preconditioner reduces
significantly the condition number of $L_k$ and performs better than the
standard incomplete Cholesky factorization.
