One of the biggest open problems in computational algebra is the design of
efficient algorithms for Gr{\"o}bner basis computations that take into account
the sparsity of the input polynomials. We can perform such computations in the
case of unmixed polynomial systems, that is systems with polynomials having the
same support, using the approach of Faug{\`e}re, Spaenlehauer, and Svartz
[ISSAC'14]. We present two algorithms for sparse Gr{\"o}bner bases computations
for mixed systems. The first one computes with mixed sparse systems and
exploits the supports of the polynomials. Under regularity assumptions, it
performs no reductions to zero. For mixed, square, and 0-dimensional
multihomogeneous polynomial systems, we present a dedicated, and potentially
more efficient, algorithm that exploits different algebraic properties that
performs no reduction to zero. We give an explicit bound for the maximal degree
appearing in the computations