  An important problem in network science is finding an optimal placement of
sensors in nodes in order to uniquely detect failures in the network. This
problem can be modelled as an identifying code set (ICS) problem, introduced by
Karpovsky et al. in 1998. The ICS problem aims to find a cover of a set $S$,
s.t. the elements in the cover define a unique signature for each of the
elements of $S$, and to minimise the cover's cardinality. In this work, we
study a generalised identifying code set (GICS) problem, where a unique
signature must be found for each subset of $S$ that has a cardinality of at
most $k$ (instead of just each element of $S$). The concept of an independent
support of a Boolean formula was introduced by Chakraborty et al. in 2014 to
speed up propositional model counting, by identifying a subset of variables
whose truth assignments uniquely define those of the other variables.
  In this work, we introduce an extended version of independent support,
grouped independent support (GIS), and show how to reduce the GICS problem to
the GIS problem. We then propose a new solving method for finding a GICS, based
on finding a GIS. We show that the prior state-of-the-art approaches yield
integer-linear programming (ILP) models whose sizes grow exponentially with the
problem size and $k$, while our GIS encoding only grows polynomially with the
problem size and $k$. While the ILP approach can solve the GICS problem on
networks of at most 494 nodes, the GIS-based method can handle networks of up
to 21363 nodes; a $\sim 40\times$ improvement. The GIS-based method shows up to
a $520\times$ improvement on the ILP-based method in terms of median solving
time. For the majority of the instances that can be encoded and solved by both
methods, the cardinality of the solution returned by the GIS-based method is
less than $10\%$ larger than the cardinality of the solution found by the ILP
method.
