  We survey research relating algebraic properties of powers of squarefree
monomial ideals to combinatorial structures. In particular, we describe how to
detect important properties of (hyper)graphs by solving ideal membership
problems and computing associated primes. This work leads to algebraic
characterizations of perfect graphs independent of the Strong Perfect Graph
Theorem. In addition, we discuss the equivalence between the
Conforti-Cornuejols conjecture from linear programming and the question of when
symbolic and ordinary powers of squarefree monomial ideals coincide.
