  We deal with the obstacle problem for the porous medium equation in the slow
diffusion regime $m>1$. Our main interest is to treat fairly irregular
obstacles assuming only boundedness and lower semicontinuity. In particular,
the considered obstacles are not regular enough to work with the classical
notion of variational solutions, and a different approach is needed. We prove
the existence of a solution in the sense of the minimal supersolution lying
above the obstacle. As a consequence, we can show that non-negative weak
supersolutions to the porous medium equation can be approximated by a sequence
of supersolutions which are bounded away from zero.
