The interleaving distance is a fundamental metric in Topological Data Analysis (TDA), with applications ranging from Reeb graphs and persistence modules to more general categorical representations of data. This framework assesses the similarity of two structures by finding a pair of parameterized maps — called an interleaving — that give a notion of an approximate isomorphism; the minimum parameter for which such an interleaving exists quantifies their distance. While the interleaving distance for 1-parameter persistence modules can be efficiently computed in polynomial time, many other cases are NP-complete — including the setting we focus on here: mapper graphs and Reeb graphs. In this talk, we introduce a new computational view of the interleaving distance for mapper graphs. We propose a loss function that measures how far a candidate structure falls short of a true interleaving. By framing the search for an interleaving as an integer linear program, we can utilize ILP solvers which apply heuristics but are often able to compute an exact answer. While the nature of ILPs mean that these results do not come with global guarantees, we show that in many cases they successfully identify the true interleaving distance. We will also demonstrate how this framework can aid in shape comparison and be naturally generalized to other TDA structures of interest.