A debt swap is an elementary edge swap in a directed, weighted graph, where two edges with the same weight swap their targets. Debt swaps are a natural and appealing operation in financial networks, in which nodes are banks and edges represent debt contracts. They can improve the clearing payments and the stability of these networks. However, their algorithmic properties are not well-understood. We analyze the computational complexity of debt swapping. Our main interest lies in semi-positive swaps, in which no creditor strictly suffers and at least one strictly profits. These swaps lead to a Pareto-improvement in the entire network. We consider network optimization via sequences of v-improving debt swaps from which a given bank v strictly profits. For ranking-based clearing, we show that every sequence of semi-positive v-improving swaps has polynomial length. In contrast, for arbitrary v-improving swaps, the problem of reaching a network configuration that allows no further swaps is PLS-complete. In global optimization, the goal is to maximize the utility of a given bank $v$ by performing a sequence of debt swaps in the network. This problem is NP-hard to approximate for multiple types of swaps. Moreover, we study reachability problems -- deciding if a sequence of swaps exists between given initial and final networks. We design a polynomial-time algorithm to decide this question for arbitrary swaps and derive hardness results for several other types of swaps. Many of our results can be extended to networks with arbitrary monotone clearing.

翻译：债务互换是有向加权图中的一种基本边交换操作，其中两条权重相同的边交换其目标节点。在金融网络中，债务互换是一种自然而引人注目的操作，其中节点代表银行，边代表债务合约。此类操作能够改善清算支付并提升网络稳定性。然而，其算法特性尚未得到充分理解。本文分析了债务互换的计算复杂性。我们的主要研究兴趣在于半正互换，即没有债权人严格受损且至少有一方严格获利的互换。这类互换能在整个网络中实现帕累托改进。我们考虑通过一系列使给定银行v严格获利的v改进债务互换来实现网络优化。对于基于排名的清算机制，我们证明所有半正v改进互换序列均具有多项式长度。相反，对于任意v改进互换，达到不存在进一步互换操作的网络配置问题是PLS完全的。在全局优化中，目标是通过执行网络中的一系列债务互换来最大化给定银行$v$的效用。对于多种互换类型，该问题的近似求解是NP难的。此外，我们研究了可达性问题——判断在给定初始网络与最终网络之间是否存在互换序列。针对任意互换类型，我们设计了多项式时间算法来判定该问题，并对其他几种互换类型推导出硬度结果。我们的许多结果可以推广到具有任意单调清算机制的网络中。