Financial networks raise a significant computational challenge in identifying insolvent firms and evaluating their exposure to systemic risk. This task, known as the clearing problem, is computationally tractable when dealing with simple debt contracts. However under the presence of certain derivatives called credit default swaps (CDSes) the clearing problem is $\textsf{FIXP}$-complete. Existing techniques only show $\textsf{PPAD}$-hardness for finding an $\epsilon$-solution for the clearing problem with CDSes within an unspecified small range for $\epsilon$. We present significant progress in both facets of the clearing problem: (i) intractability of approximate solutions; (ii) algorithms and heuristics for computable solutions. Leveraging $\textsf{Pure-Circuit}$ (FOCS'22), we provide the first explicit inapproximability bound for the clearing problem involving CDSes. Our primal contribution is a reduction from $\textsf{Pure-Circuit}$ which establishes that finding approximate solutions is $\textsf{PPAD}$-hard within a range of roughly 5%. To alleviate the complexity of the clearing problem, we identify two meaningful restrictions of the class of financial networks motivated by regulations: (i) the presence of a central clearing authority; and (ii) the restriction to covered CDSes. We provide the following results: (i.) The $\textsf{PPAD}$-hardness of approximation persists when central clearing authorities are introduced; (ii.) An optimisation-based method for solving the clearing problem with central clearing authorities; (iii.) A polynomial-time algorithm when the two restrictions hold simultaneously.

翻译：金融网络在识别资不抵债的企业及评估其系统性风险敞口方面带来了重大计算挑战。这一被称为清算问题的任务，在处理简单债务合约时具有计算可解性。然而，在存在信用违约互换（CDS）这类特定衍生品的情况下，清算问题变为$\textsf{FIXP}$-完全问题。现有技术仅能证明，在未指定的小范围$\epsilon$内，寻找带CDS的清算问题$\epsilon$-解具有$\textsf{PPAD}$-困难性。我们在清算问题的两个维度上取得了显著进展：（i）近似解的难解性；（ii）可计算解的算法与启发式方法。借助$\textsf{Pure-Circuit}$（FOCS'22），我们首次为涉及CDS的清算问题提供了显式的不可近似性界。我们的主要贡献在于从$\textsf{Pure-Circuit}$的归约，该归约证明在约5%的范围内寻找近似解是$\textsf{PPAD}$-困难的。为缓解清算问题的计算复杂性，我们基于监管要求识别出两类有意义的金融网络限制条件：（i）存在中央清算机构；（ii）限制于受覆盖CDS。我们获得以下结果：（i.）引入中央清算机构后，$\textsf{PPAD}$-难近似性仍然成立；（ii.）一种基于优化的方法用于求解带中央清算机构的清算问题；（iii.）当两个限制条件同时成立时，存在多项式时间算法。