We study the complexity of {\em rationalizing} network formation. In
this problem we fix an underlying model describing how selfish
parties (the vertices) produce a graph by making individual
decisions to form or not form incident edges. The model is equipped
with a notion of stability (or equilibrium), and we observe a set of
``snapshots'' of graphs that are assumed to be stable. From this we
would like to infer some unobserved data about the system: edge
prices, or how much each vertex values short paths to each other
vertex.

We study two rationalization problems arising from the network
formation model of Jackson and Wolinsky \cite{JW96}. When the goal
is to infer edge prices, we observe that the rationalization problem
is easy. The problem remains easy even when rationalizing prices do
not exist and we instead wish to find prices that maximize the
stability of the system.

In contrast, when the edge prices are given and the goal is instead
to infer valuations of each vertex by each other vertex, we prove
that the rationalization problem becomes NP-hard. Our proof exposes
a close connection between rationalization problems and the
Inequality-SAT (I-SAT) problem.

Finally and most significantly, we prove that an approximation
version of this NP-complete rationalization problem is NP-hard to
approximate to within better than a 1/2 ratio. This shows that the
trivial algorithm of setting everyone's valuations to infinity
(which rationalizes all the edges present in the input graphs) or to
zero (which rationalizes all the non-edges present in the input
graphs) is the best possible assuming P $\ne$ NP. To do this we
prove a tight $(1/2 + \delta)$-approximation hardness for a variant
of I-SAT in which all coefficients are non-negative. This in turn
follows from a tight hardness result for $\text{{\sc
Max-Lin}}_{{\mathbb R}_+}$ (linear equations over the reals, with
non-negative coefficients), which we prove by a (non-trivial)
modification of the recent result of Guruswami and Raghavendra
\cite{GR07} which achieved tight hardness for this problem without
the non-negativity constraint.

Our technical contributions regarding the hardness of I-SAT and
$\text{{\sc Max-Lin}}_{{\mathbb R}_+}$ may be of independent
interest, given the generality of these problems.