The network-based study of financial systems has received considerable attention in recent years but has seldom explicitly incorporated the dynamic aspects of such systems. We consider this problem setting from the temporal point of view and introduce the Interval Debt Model (IDM) and some scheduling problems based on it, namely: Bankruptcy Minimization/Maximization, in which the aim is to produce a payment schedule with at most/at least a given number of bankruptcies; Perfect Scheduling, the special case of the minimization variant where the aim is to produce a schedule with no bankruptcies (that is, a perfect schedule); and Bailout Minimization, in which a financial authority must allocate a smallest possible bailout package to enable a perfect schedule. We show that each of these problems is NP-complete, in many cases even on very restricted input instances. On the positive side, we provide for Perfect Scheduling a polynomial-time algorithm on (rooted) out-trees although in contrast we prove NP-completeness on directed acyclic graphs, as well as on instances with a constant number of nodes (and hence also constant treewidth). When we allow non-integer payments, we show by a linear programming argument that the problem Bailout Minimization can be solved in polynomial time.

翻译：基于网络的金融系统研究近年来受到广泛关注，但鲜有研究明确纳入此类系统的动态特性。我们从时间维度考量这一问题，引入区间债务模型（IDM）及其衍生的一系列调度问题，具体包括：破产最小化/最大化问题（目标在于制定支付调度方案，使得破产企业数量不超过/不低于给定值）、完美调度问题（破产最小化问题的特例，旨在实现零破产的调度方案，即完美调度），以及救助最小化问题（金融监管机构需分配尽可能小的救助方案以实现完美调度）。我们证明每个问题均为NP完全问题，即便在输入实例受到极严格限制的情况下亦成立。在积极方面，我们为（有根）外向树结构上的完美调度问题提供了多项式时间算法，但对比而言，我们证明了该问题在有向无环图上、以及节点数恒定（从而树宽恒定）的实例上均为NP完全问题。当允许非整数支付时，我们通过线性规划论证表明，救助最小化问题可在多项式时间内求解。