 
Summary: Learning a Hidden Subgraph
Noga Alon
Vera Asodi
Abstract
We consider the problem of learning a labeled graph from a given family of graphs on n vertices
in a model where the only allowed operation is to query whether a set of vertices induces an edge.
Questions of this type are motivated by problems in molecular biology. In the deterministic
nonadaptive setting, we prove nearly matching upper and lower bounds for the minimum possible
number of queries required when the family is the family of all stars of a given size or all cliques
of a given size. We further describe some bounds that apply to general graphs.
1 Introduction
Let H be a family of labeled graphs on the set V = {1, 2, . . . , n}, and suppose H is closed under
isomorphism. Given a hidden copy of some H H, we have to identify it by asking queries of the
following form. For F V , the query QF is: does F contain at least one edge of H? Our objective
is to identify H by asking as few queries as possible. We say that a family F solves the Hproblem if
for any two distinct members H1 and H2 of H, there is at least one F F that contains an edge of
one of the graphs Hi and does not contain any edge of the other. Obviously, any such family enables
us to learn an unknown member of H deterministically and nonadaptively, by asking the questions
QF for each F F. Note that for any family H, the set of all pairs of vertices solves the Hproblem.
Note also that the information theoretic lower bound implies that we need at least log H queries,
