Payment Scheduling in the Interval Debt Model

TL;DR

This work introduces the Interval Debt Model (IDM), a temporal network framework where debts have explicit payment windows, and studies scheduling problems such as Bankruptcy Minimization, Perfect Scheduling, and Bailout Minimization across AoN, PP, and FP variants. It establishes NP-completeness for many problem variants, even on highly restricted graphs, while identifying tractable cases: Perfect Scheduling on rooted out-trees, and Bailout Minimization solvable in polynomial time for FP via linear programming and for PP on out-trees via graph-transformations. The results illuminate how temporal constraints and partial/fractional payments affect computational complexity, offering both hard instances and efficient algorithms in structurally restricted settings. The findings have potential implications for regulatory policies and the design of bailout strategies in financial networks, highlighting where centralized control can achieve efficient risk mitigation and where intractability arises despite structural simplifications.

Abstract

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.

Figures (18)

• Figure 1: A simple instance of the Interval Debt Model (IDM). Numbers in square boxes represent the initial external assets of the node (for example, €30 for node $u$), directed edges represent debts, and the label on an edge represents the terms of the associated debt (for example, $u$ must pay $v$€20 between time 1 and time 3).
• Figure 2: An IDM instance for which every schedule is described by four payment values $p_{(u,v,0)}^1$, $p_{(v,w,0)}^1$, $p_{(u,v,0)}^2$ and $p_{(v,w,0)}^2$.
• Figure 3: Examples illustrating the behaviour of cycles in the IDM. In all instances shown the schedule in which all nodes pay their debts in full at time 1 is valid.
• Figure 4: Using a payment-cycle to effectively transfer €1 of assets from node $u$ to node $v$.
• Figure 5: A real-life interval debt: this 1978 US government bond is payable between 2003 and 2008.

Theorems & Definitions (69)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Lemma 1
• proof : Proof sketch
• Theorem 1
• proof
• Claim 1