  We introduce Selective Greedy Equivalence Search (SGES), a restricted version
of Greedy Equivalence Search (GES). SGES retains the asymptotic correctness of
GES but, unlike GES, has polynomial performance guarantees. In particular, we
show that when data are sampled independently from a distribution that is
perfect with respect to a DAG ${\cal G}$ defined over the observable variables
then, in the limit of large data, SGES will identify ${\cal G}$'s equivalence
class after a number of score evaluations that is (1) polynomial in the number
of nodes and (2) exponential in various complexity measures including
maximum-number-of-parents, maximum-clique-size, and a new measure called {\em
v-width} that is at least as small as---and potentially much smaller than---the
other two. More generally, we show that for any hereditary and
equivalence-invariant property $\Pi$ known to hold in ${\cal G}$, we retain the
large-sample optimality guarantees of GES even if we ignore any GES deletion
operator during the backward phase that results in a state for which $\Pi$ does
not hold in the common-descendants subgraph.
