We study financial networks where banks are connected through bilateral liabilities and may default when resources are insufficient to meet obligations. We consider both the standard proportional clearing model and a priority-proportional clearing model in which banks repay creditors according to exogenously given priority classes. In such markets, portfolio compression is a process where several banks come to a netting arrangement which reduces liabilities without changing any bank's net exposure, essentially removing cycles of debt. Our goal is to understand whether portfolio compression schemes can be designed to improve clearing outcomes for a large fraction of banks. We provide a computational characterization of the benefits and limitations of compression. On the positive side, we give a polynomial-time algorithm to compute a maximal clearing outcome under priority-proportional clearing, and we show that it is possible to decide in polynomial time whether there exists a compression that limits defaults to at most one bank. On the negative side, we show that several natural optimization and decision problems are computationally intractable: deciding whether some compression can reduce the number of defaulting banks below a given threshold, or whether a specific bank can be saved from defaulting, is $\NP$-hard even in restricted settings and under proportional clearing. We further present a mixed integer linear programming (MILP) formulation that computes a compression maximizing the number of non-defaulting banks, providing a practical approach to this hard problem. Using our MILP formulation, we perform simulations on both synthetic and real-world datasets to analyze the effects of portfolio compression.

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