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公司并评估其系统性风险敞口时，带来了重大的计算挑战。这一任务被称为清算问题，在处理简单债务合约时在计算上是可解的。然而，在存在某些名为信用违约互换（CDS）的衍生品时，清算问题是$\textsf{FIXP}$-完全的。现有技术仅表明，在未指定的微小$\epsilon$范围内，寻找带CDS的清算问题的$\epsilon$-解是$\textsf{PPAD}$-难的。我们在清算问题的两个方面取得了显著进展：(i) 近似解难解性；(ii) 可计算解的算法与启发式方法。利用$\textsf{Pure-Circuit}$（FOCS'22），我们为涉及CDS的清算问题提供了首个显式不可近似性界。我们的主要贡献是从$\textsf{Pure-Circuit}$的归约，该归约表明在大约5%的范围内寻找近似解是$\textsf{PPAD}$-难的。为缓解清算问题的复杂性，我们识别了受监管启发的两类有意义的金融网络限制：(i) 存在中央清算机构；(ii) 仅限于担保型CDS。我们得到以下结果：(i.) 引入中央清算机构后，$\textsf{PPAD}$-难近似性仍然存在；(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.) 当两类限制同时成立时，存在一个多项式时间算法。