Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Fateme Abbasi, Marek Adamczyk, Miguel Bosch-Calvo, Jarosław Byrka, Fabrizio Grandoni, Krzysztof Sornat, and Antoine Tinguely. An O(loglog n)-Approximation for Submodular Facility Location. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{abbasi_et_al:LIPIcs.ICALP.2024.5, author = {Abbasi, Fateme and Adamczyk, Marek and Bosch-Calvo, Miguel and Byrka, Jaros{\l}aw and Grandoni, Fabrizio and Sornat, Krzysztof and Tinguely, Antoine}, title = {{An O(loglog n)-Approximation for Submodular Facility Location}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {5:1--5:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.5}, URN = {urn:nbn:de:0030-drops-201488}, doi = {10.4230/LIPIcs.ICALP.2024.5}, annote = {Keywords: approximation algorithms, facility location, submodular facility location, universal stochastic facility location} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. Lower Bounds for Matroid Optimization Problems with a Linear Constraint. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{doronarad_et_al:LIPIcs.ICALP.2024.56, author = {Doron-Arad, Ilan and Kulik, Ariel and Shachnai, Hadas}, title = {{Lower Bounds for Matroid Optimization Problems with a Linear Constraint}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {56:1--56:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.56}, URN = {urn:nbn:de:0030-drops-201990}, doi = {10.4230/LIPIcs.ICALP.2024.56}, annote = {Keywords: matroid optimization, budgeted problems, knapsack, approximation schemes} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Yuda Feng and Shi Li. A Note on Approximating Weighted Nash Social Welfare with Additive Valuations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 63:1-63:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{feng_et_al:LIPIcs.ICALP.2024.63, author = {Feng, Yuda and Li, Shi}, title = {{A Note on Approximating Weighted Nash Social Welfare with Additive Valuations}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {63:1--63:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.63}, URN = {urn:nbn:de:0030-drops-202068}, doi = {10.4230/LIPIcs.ICALP.2024.63}, annote = {Keywords: Nash Social Welfare, Configuration LP, Approximation Algorithms} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Yotam Kenneth and Robert Krauthgamer. Cut Sparsification and Succinct Representation of Submodular Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 97:1-97:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{kenneth_et_al:LIPIcs.ICALP.2024.97, author = {Kenneth, Yotam and Krauthgamer, Robert}, title = {{Cut Sparsification and Succinct Representation of Submodular Hypergraphs}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {97:1--97:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.97}, URN = {urn:nbn:de:0030-drops-202406}, doi = {10.4230/LIPIcs.ICALP.2024.97}, annote = {Keywords: Cut Sparsification, Submodular Hypergraphs, Succinct Representation} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Yusuke Kobayashi and Tatsuya Terao. Subquadratic Submodular Maximization with a General Matroid Constraint. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 100:1-100:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{kobayashi_et_al:LIPIcs.ICALP.2024.100, author = {Kobayashi, Yusuke and Terao, Tatsuya}, title = {{Subquadratic Submodular Maximization with a General Matroid Constraint}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {100:1--100:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.100}, URN = {urn:nbn:de:0030-drops-202437}, doi = {10.4230/LIPIcs.ICALP.2024.100}, annote = {Keywords: submodular maximization, matroid constraint, approximation algorithm, rounding algorithm, query complexity} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Shi Li, Chenyang Xu, and Ruilong Zhang. Polylogarithmic Approximations for Robust s-t Path. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 106:1-106:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{li_et_al:LIPIcs.ICALP.2024.106, author = {Li, Shi and Xu, Chenyang and Zhang, Ruilong}, title = {{Polylogarithmic Approximations for Robust s-t Path}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {106:1--106:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.106}, URN = {urn:nbn:de:0030-drops-202497}, doi = {10.4230/LIPIcs.ICALP.2024.106}, annote = {Keywords: Approximation Algorithm, Randomized LP Rounding, Robust s-t Path} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Abhishek Sahu, Saket Saurabh, and Anannya Upasana. Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 88:1-88:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{inamdar_et_al:LIPIcs.ICALP.2024.88, author = {Inamdar, Tanmay and Jain, Pallavi and Lokshtanov, Daniel and Sahu, Abhishek and Saurabh, Saket and Upasana, Anannya}, title = {{Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {88:1--88:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.88}, URN = {urn:nbn:de:0030-drops-202318}, doi = {10.4230/LIPIcs.ICALP.2024.88}, annote = {Keywords: Partial Vertex Cover, Max SAT, FPT Approximation, Matroids} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Yury Makarychev, Max Ovsiankin, and Erasmo Tani. Approximation Algorithms for 𝓁_p-Shortest Path and 𝓁_p-Group Steiner Tree. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 111:1-111:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{makarychev_et_al:LIPIcs.ICALP.2024.111, author = {Makarychev, Yury and Ovsiankin, Max and Tani, Erasmo}, title = {{Approximation Algorithms for 𝓁\underlinep-Shortest Path and 𝓁\underlinep-Group Steiner Tree}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {111:1--111:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.111}, URN = {urn:nbn:de:0030-drops-202542}, doi = {10.4230/LIPIcs.ICALP.2024.111}, annote = {Keywords: Shortest Path, Asymmetric Group Steiner Tree, Sum-of-Squares} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Orestis Plevrakis, Seyoon Ragavan, and S. Matthew Weinberg. On the Cut-Query Complexity of Approximating Max-Cut. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{plevrakis_et_al:LIPIcs.ICALP.2024.115, author = {Plevrakis, Orestis and Ragavan, Seyoon and Weinberg, S. Matthew}, title = {{On the Cut-Query Complexity of Approximating Max-Cut}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {115:1--115:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.115}, URN = {urn:nbn:de:0030-drops-202587}, doi = {10.4230/LIPIcs.ICALP.2024.115}, annote = {Keywords: query complexity, maximum cut, approximation algorithms, graph sparsification} }
Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Monika Henzinger, Paul Liu, Jan Vondrák, and Da Wei Zheng. Faster Submodular Maximization for Several Classes of Matroids. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 74:1-74:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
@InProceedings{henzinger_et_al:LIPIcs.ICALP.2023.74, author = {Henzinger, Monika and Liu, Paul and Vondr\'{a}k, Jan and Zheng, Da Wei}, title = {{Faster Submodular Maximization for Several Classes of Matroids}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {74:1--74:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.74}, URN = {urn:nbn:de:0030-drops-181267}, doi = {10.4230/LIPIcs.ICALP.2023.74}, annote = {Keywords: submodular optimization, dynamic data structures, matching algorithms} }
Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Moran Feldman, Paul Liu, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen. Streaming Submodular Maximization Under Matroid Constraints. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 59:1-59:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
@InProceedings{feldman_et_al:LIPIcs.ICALP.2022.59, author = {Feldman, Moran and Liu, Paul and Norouzi-Fard, Ashkan and Svensson, Ola and Zenklusen, Rico}, title = {{Streaming Submodular Maximization Under Matroid Constraints}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {59:1--59:20}, 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.2022.59}, URN = {urn:nbn:de:0030-drops-164007}, doi = {10.4230/LIPIcs.ICALP.2022.59}, annote = {Keywords: Submodular maximization, streaming, matroid, random order} }
Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
Linda Cai, Clay Thomas, and S. Matthew Weinberg. Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 61:1-61:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
@InProceedings{cai_et_al:LIPIcs.ITCS.2020.61, author = {Cai, Linda and Thomas, Clay and Weinberg, S. Matthew}, title = {{Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {61:1--61:32}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.61}, URN = {urn:nbn:de:0030-drops-117464}, doi = {10.4230/LIPIcs.ITCS.2020.61}, annote = {Keywords: Combinatorial auctions, Posted-Price mechanisms, Submodular valuations, Incentive compatible} }
Published in: OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)
Paul Liu and Jan Vondrak. Submodular Optimization in the MapReduce Model. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 18:1-18:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
@InProceedings{liu_et_al:OASIcs.SOSA.2019.18, author = {Liu, Paul and Vondrak, Jan}, title = {{Submodular Optimization in the MapReduce Model}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {18:1--18:10}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.18}, URN = {urn:nbn:de:0030-drops-100447}, doi = {10.4230/OASIcs.SOSA.2019.18}, annote = {Keywords: mapreduce, submodular, optimization, approximation algorithms} }
Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)
Vaggos Chatziafratis, Tim Roughgarden, and Jan Vondrak. Stability and Recovery for Independence Systems. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
@InProceedings{chatziafratis_et_al:LIPIcs.ESA.2017.26, author = {Chatziafratis, Vaggos and Roughgarden, Tim and Vondrak, Jan}, title = {{Stability and Recovery for Independence Systems}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {26:1--26:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.26}, URN = {urn:nbn:de:0030-drops-78423}, doi = {10.4230/LIPIcs.ESA.2017.26}, annote = {Keywords: Submodular, approximation, stability, Local Search, Greedy, p-systems} }
Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)
Tim Roughgarden, Inbal Talgam-Cohen, and Jan Vondrák. When Are Welfare Guarantees Robust?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
@InProceedings{roughgarden_et_al:LIPIcs.APPROX-RANDOM.2017.22, author = {Roughgarden, Tim and Talgam-Cohen, Inbal and Vondr\'{a}k, Jan}, title = {{When Are Welfare Guarantees Robust?}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {22:1--22:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.22}, URN = {urn:nbn:de:0030-drops-75714}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.22}, annote = {Keywords: Valuation (set) functions, gross substitutes, linearity, approximation} }