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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

The recent banking crisis has again emphasized the importance of understanding and mitigating systemic risk in financial networks. In this paper, we study a market-driven approach to rescue a bank in distress based on the idea of claims trading, a notion defined in Chapter 11 of the U.S. Bankruptcy Code. We formalize the idea in the context of the seminal model of financial networks by Eisenberg and Noe [Eisenberg and Noe, 2001]. For two given banks v and w, we consider the operation that w takes over some claims of v and in return gives liquidity to v (or creditors of v) to ultimately rescue v (or mitigate contagion effects). We study the structural properties and computational complexity of decision and optimization problems for several variants of claims trading.
When trading incoming edges of v (i.e., claims for which v is the creditor), we show that there is no trade in which both banks v and w strictly improve their assets. We therefore consider creditor-positive trades, in which v profits strictly and w remains indifferent. For a given set C of incoming edges of v, we provide an efficient algorithm to compute payments by w that result in a creditor-positive trade and maximal assets of v. When the set C must also be chosen, the problem becomes weakly NP-hard. Our main result here is a bicriteria FPTAS to compute an approximate trade, which allows for slightly increased payments by w. The approximate trade results in nearly the optimal amount of assets of v in any exact trade. Our results extend to the case in which banks use general monotone payment functions to settle their debt and the emerging clearing state can be computed efficiently.
In contrast, for trading outgoing edges of v (i.e., claims for which v is the debtor), the goal is to maximize the increase in assets for the creditors of v. Notably, for these results the characteristics of the payment functions of the banks are essential. For payments ranking creditors one by one, we show NP-hardness of approximation within a factor polynomial in the network size, in both problem variants when the set of claims C is part of the input or not. Instead, for payments proportional to the value of each debt, our results indicate more favorable conditions.

Martin Hoefer, Carmine Ventre, and Lisa Wilhelmi. Algorithms for Claims Trading. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hoefer_et_al:LIPIcs.STACS.2024.42, author = {Hoefer, Martin and Ventre, Carmine and Wilhelmi, Lisa}, title = {{Algorithms for Claims Trading}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {42:1--42:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.42}, URN = {urn:nbn:de:0030-drops-197523}, doi = {10.4230/LIPIcs.STACS.2024.42}, annote = {Keywords: Financial Networks, Claims Trade, Systemic Risk} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least 0.5721 - o(1) for growing n, and an algorithm with ratio at least 0.5459 for all n >= 1. We extend all algorithms and ratios to k >= 2 arrivals per candidate.
In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Theta(n log n) is always sufficient. For matroids, we show that the expected number can be reduced to O(r log (n/r)), where r <=n/2 is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of O(r log n), where r is the size of the optimum matching. For general packing, we show a lower bound of Omega(n log log n), even when the size of the optimum is r = Theta(log n).

Martin Hoefer and Lisa Wilhelmi. Packing Returning Secretaries. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 65:1-65:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hoefer_et_al:LIPIcs.ISAAC.2018.65, author = {Hoefer, Martin and Wilhelmi, Lisa}, title = {{Packing Returning Secretaries}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {65:1--65:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.65}, URN = {urn:nbn:de:0030-drops-100133}, doi = {10.4230/LIPIcs.ISAAC.2018.65}, annote = {Keywords: Secretary Problem, Coupon Collector Problem, Matroids} }

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