Modern financial networks are highly connected and result in complex interdependencies of the involved institutions. In the prominent Eisenberg-Noe model, a fundamental aspect is clearing -- to determine the amount of assets available to each financial institution in the presence of potential defaults and bankruptcy. A clearing state represents a fixed point that satisfies a set of natural axioms. Existence can be established (even in broad generalizations of the model) using Tarski's theorem. While a maximal fixed point can be computed in polynomial time, the complexity of computing other fixed points is open. In this paper, we provide an efficient algorithm to compute a minimal fixed point that runs in strongly polynomial time. It applies in a broad generalization of the Eisenberg-Noe model with any monotone, piecewise-linear payment functions and default costs. Moreover, in this scenario we provide a polynomial-time algorithm to compute a maximal fixed point. For networks without default costs, we can efficiently decide the existence of fixed points in a given range. We also study claims trading, a local network adjustment to improve clearing, when networks are evaluated with minimal clearing. We provide an efficient algorithm to decide existence of Pareto-improving trades and compute optimal ones if they exist.

翻译：现代金融网络高度互联，导致相关机构之间存在复杂的相互依赖性。在著名的艾森伯格-诺埃模型中，一个基本方面是清算——在存在潜在违约和破产的情况下，确定每个金融机构可用的资产数量。清算状态代表满足一组自然公理的不动点。使用塔尔斯基定理可以证明其存在性（即使在模型的广泛推广中）。虽然最大不动点可以在多项式时间内计算，但计算其他不动点的复杂度尚未解决。本文提出了一种在强多项式时间内计算最小不动点的高效算法。该算法适用于具有任意单调分段线性支付函数和违约成本的艾森伯格-诺埃模型的广泛推广。此外，在此场景下，我们提供了一种计算最大不动点的多项式时间算法。对于无违约成本的网络，我们可以高效判定给定范围内不动点的存在性。我们还研究了当网络采用最小清算评估时，索赔交易这种旨在改善清算的局部网络调整机制。我们提出了一种高效算法来判定帕累托改进交易的存在性，并在存在时计算最优交易。