金融网络中的动态债务交换
Dynamic Debt Swapping in Financial Networks
AI总结：
研究金融网络中债务交换的算法性质，分析半正交换的计算复杂性及全局优化问题。
摘要：
债务交换是有向加权图中的基本边交换操作，其中两条权重相同的边交换它们的目标。债务交换是金融网络中自然且有吸引力的操作，其中节点是银行，边代表债务合同。它们可以提高清算支付和网络的稳定性。然而，其算法性质尚不明确。我们分析了债务交换的计算复杂性。我们的主要兴趣在于半正交换，其中没有债权人严格受损且至少有一个严格获利。这些交换导致整个网络的帕累托改进。我们考虑通过一系列v-改进的债务交换进行网络优化，其中给定的银行v严格获利。对于基于排名的清算，我们证明了每一条半正v-改进交换序列都有多项式长度。相比之下，对于任意v-改进交换，达到一个不允许进一步交换的网络配置的问题是PLS完全的。在全局优化中，目标是通过在网络中进行一系列债务交换来最大化给定银行v的效用。这个问题对于多种类型的交换都是NP难的。此外，我们研究了可达性问题——决定给定初始和最终网络之间是否存在交换序列。我们为任意交换设计了多项式时间算法来决定这个问题，并推导了其他几种交换类型的硬度结果。许多我们的结果可以扩展到具有任意单调清算的网络。
英文摘要：
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.
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