  In this paper we study a resource allocation problem that encodes correlation
between items in terms of \conflict and maximizes the minimum utility of the
agents under a conflict free allocation. Admittedly, the problem is
computationally hard even under stringent restrictions because it encodes a
variant of the {\sc Maximum Weight Independent Set} problem which is one of the
canonical hard problems in both classical and parameterized complexity.
Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from
the classical complexity perspective to draw the boundary between {\sf
NP}-hardness and tractability for a constant number of agents. The problem was
shown to be hard even for small constant number of agents and various other
restrictions on the underlying graph. Notwithstanding this computational
barrier, we notice that there are several parameters that are worth studying:
number of agents, number of items, combinatorial structure that defines the
conflict among the items, all of which could well be small under specific
circumstancs. Our search rules out several parameters (even when taken
together) and takes us towards a characterization of families of input
instances that are amenable to polynomial time algorithms when the parameters
are constant. In addition to this we give a superior $2^{m}|I|^{\Co{O}(1)}$
algorithm for our problem where $m$ denotes the number of items that
significantly beats the exhaustive $\Oh(m^{m})$ algorithm by cleverly using
ideas from FFT based fast polynomial multiplication; and we identify simple
graph classes relevant to our problem's motivation that admit efficient
algorithms.
