Payment scheduling in the Interval Debt Model

Abstract

The networks-based study of financial systems has received considerable attention in recent years, but seldom explicitly incorporated the dynamic aspects of such systems. We consider this problem setting from the temporal point of view, and we 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 schedule with at most / at least k bankruptcies; Perfect Scheduling, the special case of the minimization variant where k=0; and Bailout Minimization, in which a financial authority must allocate a smallest possible bailout package to enable a perfect schedule. In this paper we investigate the complexity landscape of the various variants of these problems. We show that each of them is NP-complete, in many cases even on very restrictive input instances. On the positive side, we provide for Perfect Scheduling a polynomial-time algorithm on (rooted) out-trees. In wide contrast, we prove that this problem is NP-complete on directed acyclic graphs (DAGs), as well as on instances with a constant number of nodes (and hence also constant treewidth). When the problem definition is relaxed to allow fractional payments, we show by a linear programming argument that Bailout Minimization can be solved in polynomial time.