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 in networks with ranking-based clearing. 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. 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. We identify cases in which short sequences of semi-positive swaps exist even without the $v$-improving property. In addition, we study reachability problems, i.e., deciding if a sequence of swaps exists between given initial and final networks. We identify a polynomial-time algorithm for arbitrary swaps, show NP-hardness for semi-positive swaps and even PSPACE-completeness for $v$-improving swaps or swaps that only maintain a lower bound on the assets of a given bank $v$. A variety of our results can be extended to arbitrary monotone clearing.

翻译：债务交换是一种在有向加权图中进行的边交换操作，即两条权重相同的边交换它们的目标节点。在金融网络中，债务交换是一种自然且引人关注的操作，其中节点代表银行，边代表债务合同。这种操作能够改善清算支付并提升网络的稳定性，但其算法特性尚未得到充分理解。我们分析了基于排名清算的网络中债务交换的计算复杂性。我们的主要关注点是半正交换，即没有债权人遭受严格损失且至少有一个债权人获得严格收益的交换。这类交换能在整个网络中实现帕累托改进。我们考虑了通过一系列使给定银行$v$获得严格收益的$v$改进债务交换来实现网络优化。我们证明，每轮半正$v$改进交换序列的长度都是多项式级别的。相比之下，对于任意$v$改进交换，达到不允许进一步交换的网络配置问题是PLS完全的。我们还识别出了一些即使不满足$v$改进性质也能存在较短半正交换序列的情况。此外，我们研究了可达性问题，即判断在给定的初始网络和最终网络之间是否存在一系列交换。我们为任意交换找到了一个多项式时间算法，证明了半正交换的NP困难性，以及$v$改进交换或仅维持给定银行$v$资产下限的交换的PSPACE完全性。我们的多项结果可以推广到任意单调清算场景。