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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Financial networks model a set of financial institutions (firms) interconnected by obligations. Recent work has introduced to this model a class of obligations called credit default swaps, a certain kind of financial derivatives. The main computational challenge for such systems is known as the clearing problem, which is to determine which firms are in default and to compute their exposure to systemic risk, technically known as their recovery rates. It is known that the recovery rates form the set of fixed points of a simple function, and that these fixed points can be irrational. Furthermore, Schuldenzucker et al. (2016) have shown that finding a weakly (or "almost") approximate (rational) fixed point is PPAD-complete.
We further study the clearing problem from the point of view of irrationality and approximation strength. Firstly, we observe that weakly approximate solutions may misrepresent the actual financial state of an institution. On this basis, we study the complexity of finding a strongly (or "near") approximate solution, and show FIXP-completeness. We then study the structural properties required for irrationality, and we give necessary conditions for irrational solutions to emerge: The presence of certain types of cycles in a financial network forces the recovery rates to take the form of roots of non-linear polynomials. In the absence of a large subclass of such cycles, we study the complexity of finding an exact fixed point, which we show to be a problem close to, albeit outside of, PPAD.

Stavros D. Ioannidis, Bart de Keijzer, and Carmine Ventre. Strong Approximations and Irrationality in Financial Networks with Derivatives. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 76:1-76:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ioannidis_et_al:LIPIcs.ICALP.2022.76, author = {Ioannidis, Stavros D. and de Keijzer, Bart and Ventre, Carmine}, title = {{Strong Approximations and Irrationality in Financial Networks with Derivatives}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {76:1--76:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.76}, URN = {urn:nbn:de:0030-drops-164172}, doi = {10.4230/LIPIcs.ICALP.2022.76}, annote = {Keywords: FIXP, Financial Networks, Systemic Risk} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We consider the setting of combinatorial auctions when the agents are single-minded and have no contingent reasoning skills. We are interested in mechanisms that provide the right incentives to these imperfectly rational agents, and therefore focus our attention to obviously strategyproof (OSP) mechanisms. These mechanisms require that at each point during the execution where an agent is queried to communicate information, it should be "obvious" for the agent what strategy to adopt in order to maximise her utility. In this paper we study the potential of OSP mechanisms with respect to the approximability of the optimal social welfare.
We consider two cases depending on whether the desired bundles of the agents are known or unknown to the mechanism. For the case of known-bundle single-minded agents we show that OSP can actually be as powerful as (plain) strategyproofness (SP). In particular, we show that we can implement the very same algorithm used for SP to achieve a √m-approximation of the optimal social welfare with an OSP mechanism, m being the total number of items. Restricting our attention to declaration domains with two values, we provide a 2-approximate OSP mechanism, and prove that this approximation bound is tight. We also present a randomised mechanism that is universally OSP and achieves a finite approximation of the optimal social welfare for the case of arbitrary size finite domains. This mechanism also provides a bounded approximation ratio when the valuations lie in a bounded interval (even if the declaration domain is infinitely large). For the case of unknown-bundle single-minded agents, we show how we can achieve an approximation ratio equal to the size of the largest desired set, in an OSP way. We remark this is the first known application of OSP to multi-dimensional settings, i.e., settings where agents have to declare more than one parameter.
Our results paint a rather positive picture regarding the power of OSP mechanisms in this context, particularly for known-bundle single-minded agents. All our results are constructive, and even though some known strategyproof algorithms are used, implementing them in an OSP way is a non-trivial task.

Bart de Keijzer, Maria Kyropoulou, and Carmine Ventre. Obviously Strategyproof Single-Minded Combinatorial Auctions. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 71:1-71:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dekeijzer_et_al:LIPIcs.ICALP.2020.71, author = {de Keijzer, Bart and Kyropoulou, Maria and Ventre, Carmine}, title = {{Obviously Strategyproof Single-Minded Combinatorial Auctions}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {71:1--71:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.71}, URN = {urn:nbn:de:0030-drops-124781}, doi = {10.4230/LIPIcs.ICALP.2020.71}, annote = {Keywords: OSP Mechanisms, Extensive-form Mechanisms, Single-minded Combinatorial Auctions, Greedy algorithms} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph G = (V, E), where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges T subseteq E such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. Besides being a natural generalization of the well-studied maximum coverage problem, our motivation for investigating this problem originates from its application in the context of bid optimization in sponsored search auctions, such as Google AdWords.
It is easily seen that this problem is strictly harder than budgeted max coverage, which means that the problem is (1-1/e)-inapproximable. The difference of our problem to the budgeted maximum coverage problem is that the costs are associated with the covered vertices instead of the selected hyperedges. As it turns out, this difference refutes the applicability of standard greedy approaches which are used to obtain constant factor approximation algorithms for several other variants of the maximum coverage problem. Our main results are as follows:
- We obtain a (1 - 1/sqrt(e))/2-approximation algorithm for graphs.
- We derive a fully polynomial-time approximation scheme (FPTAS) if the incidence graph of the hypergraph is a forest (i.e., the hypergraph is Berge-acyclic). We also extend this result to incidence graphs with a fixed-size feedback hyperedge node set.
- We give a (1-epsilon)/(2d^2)-approximation algorithm for every epsilon > 0, where d is the maximum degree of a vertex in the hypergraph.

Irving van Heuven van Staereling, Bart de Keijzer, and Guido Schäfer. The Ground-Set-Cost Budgeted Maximum Coverage Problem. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{vanheuvenvanstaereling_et_al:LIPIcs.MFCS.2016.50, author = {van Heuven van Staereling, Irving and de Keijzer, Bart and Sch\"{a}fer, Guido}, title = {{The Ground-Set-Cost Budgeted Maximum Coverage Problem}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {50:1--50:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.50}, URN = {urn:nbn:de:0030-drops-65020}, doi = {10.4230/LIPIcs.MFCS.2016.50}, annote = {Keywords: maximum coverage problem, approximation algorithms, hypergraphs, submodular optimization, sponsored search} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.

Haris Aziz and Bart de Keijzer. Shapley meets Shapley. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 99-111, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{aziz_et_al:LIPIcs.STACS.2014.99, author = {Aziz, Haris and de Keijzer, Bart}, title = {{Shapley meets Shapley}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {99--111}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.99}, URN = {urn:nbn:de:0030-drops-44504}, doi = {10.4230/LIPIcs.STACS.2014.99}, annote = {Keywords: matching games, Shapley, counting complexity} }