{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,4]],"date-time":"2025-05-04T04:02:27Z","timestamp":1746331347628,"version":"3.40.4"},"publisher-location":"Singapore","reference-count":23,"publisher":"Springer Nature Singapore","isbn-type":[{"value":"9789819644476","type":"print"},{"value":"9789819644483","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025]]},"DOI":"10.1007\/978-981-96-4448-3_9","type":"book-chapter","created":{"date-parts":[[2025,5,3]],"date-time":"2025-05-03T05:50:50Z","timestamp":1746251450000},"page":"107-118","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Computing Approximate Mixed Nash Equilibria for\u00a0Symmetric Weighted Congestion Games"],"prefix":"10.1007","author":[{"given":"Chunying","family":"Ren","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zijun","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoguang","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guoqing","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,4]]},"reference":[{"issue":"1","key":"9_CR1","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF01737559","volume":"2","author":"RW Rosenthal","year":"1973","unstructured":"Rosenthal, R.W.: A class of games possessing pure-strategy nash equilibria. Internat. J. Game Theory 2(1), 65\u201367 (1973)","journal-title":"Internat. J. Game Theory"},{"key":"9_CR2","doi-asserted-by":"crossref","unstructured":"Dafermos, S.C., Sparrow, F.T.: The traffic assignment problem for a general network. J. Res. U.S. Natl. Bureau Stand. 73(2), 91\u2013118 (1969)","DOI":"10.6028\/jres.073B.010"},{"key":"9_CR3","doi-asserted-by":"crossref","unstructured":"Nisan, N., Roughgarden, T., Tardos, $$\\acute{E}$$., Vazirani, V.: Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)","DOI":"10.1017\/CBO9780511800481"},{"issue":"6","key":"9_CR4","doi-asserted-by":"publisher","first-page":"734","DOI":"10.26599\/TST.2020.9010005","volume":"25","author":"X Song","year":"2020","unstructured":"Song, X., Jiang, W., Liu, X., Lu, H., Tian, Z., Du, X.: A survey of game theory as applied to social networks. Tsinghua Sci. Technol. 25(6), 734\u2013742 (2020)","journal-title":"Tsinghua Sci. Technol."},{"issue":"2","key":"9_CR5","doi-asserted-by":"publisher","first-page":"236","DOI":"10.1145\/506147.506153","volume":"49","author":"T Roughgarden","year":"2002","unstructured":"Roughgarden, T., Tardos, \u00c9.: How bad is selfish routing? J. ACM 49(2), 236\u2013259 (2002)","journal-title":"J. ACM"},{"issue":"4","key":"9_CR6","doi-asserted-by":"publisher","first-page":"961","DOI":"10.1287\/moor.1040.0098","volume":"29","author":"JR Correa","year":"2004","unstructured":"Correa, J.R., Schulz, A.S., Moses, N.: Selfish routing in capacitated networks. Math. Oper. Res. 29(4), 961\u2013976 (2004)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"9_CR7","first-page":"411","volume":"68","author":"R Colini-Baldeschi","year":"2020","unstructured":"Colini-Baldeschi, R., Cominetti, R., Mertikopoulos, P., Scarsini, M.: When is selfish routing bad? the price of anarchy in light and heavy traffic. Oper. Res. 68(2), 411\u2013434 (2020)","journal-title":"Oper. Res."},{"issue":"2","key":"9_CR8","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1287\/opre.2020.2036","volume":"69","author":"Z Wu","year":"2021","unstructured":"Wu, Z., M\u00f6hring, R., Chen, Y., Xu, D.: Selfishness need not be bad. Oper. Res. 69(2), 410\u2013435 (2021)","journal-title":"Oper. Res."},{"issue":"3","key":"9_CR9","doi-asserted-by":"publisher","first-page":"1364","DOI":"10.1287\/moor.2022.1292","volume":"48","author":"Z Wu","year":"2023","unstructured":"Wu, Z., M\u00f6hring, R.: A sensitivity analysis of the price of anarchy in non-atomic congestion games. Math. Oper. Res. 48(3), 1364\u20131392 (2023)","journal-title":"Math. Oper. Res."},{"issue":"1","key":"9_CR10","doi-asserted-by":"publisher","first-page":"937","DOI":"10.1007\/s10107-022-01853-0","volume":"199","author":"Z Wu","year":"2023","unstructured":"Wu, Z., M\u00f6hring, R., Ren, C., Xu, D.: A convergence analysis of the price of anarchy in atomic congestion games. Math. Program. 199(1), 937\u2013993 (2023)","journal-title":"Math. Program."},{"issue":"2","key":"9_CR11","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1016\/S1007-0214(07)70027-5","volume":"12","author":"H Lu","year":"2007","unstructured":"Lu, H., Shi, Y.: Complexity of public transport networks. Tsinghua Sci. Technol. 12(2), 204\u2013213 (2007)","journal-title":"Tsinghua Sci. Technol."},{"key":"9_CR12","doi-asserted-by":"crossref","unstructured":"Nash, J.F.: Equilibrium points in $$n$$-person games. In: Proceedings of the National Academy of Sciences, vol. 36, pp. 48\u201349 (1950)","DOI":"10.1073\/pnas.36.1.48"},{"issue":"3","key":"9_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1516512.1516516","volume":"56","author":"X Chen","year":"2009","unstructured":"Chen, X., Deng, X., Teng, S.H.: Settling the complexity of computing two-player nash equilibria. J. ACM (JACM) 56(3), 1\u201357 (2009)","journal-title":"J. ACM (JACM)"},{"key":"9_CR14","doi-asserted-by":"crossref","unstructured":"Papadimitriou, C.H.: On inefficient proofs of existence and complexity classes. In: Proceedings of the 4th Czechoslovakian Symposium on Combinatorics (1991)","DOI":"10.1016\/S0167-5060(08)70637-X"},{"issue":"2\u20133","key":"9_CR15","doi-asserted-by":"publisher","first-page":"226","DOI":"10.1016\/j.tcs.2005.09.024","volume":"348","author":"D Fotakis","year":"2005","unstructured":"Fotakis, D., Kontogiannis, S., Spirakis, P.G.: Selfish unsplittable flows. Theoret. Comput. Sci. 348(2\u20133), 226\u2013239 (2005)","journal-title":"Theoret. Comput. Sci."},{"key":"9_CR16","doi-asserted-by":"crossref","unstructured":"Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 604\u2013612. ACM (2004)","DOI":"10.1145\/1007352.1007445"},{"issue":"4","key":"9_CR17","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1287\/moor.1080.0322","volume":"33","author":"J Dunkel","year":"2008","unstructured":"Dunkel, J., Schulz, A.S.: On the complexity of pure-strategy nash equilibria in congestion and local effect games. Math. Oper. Res. 33(4), 851\u2013868 (2008)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"9_CR18","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1016\/j.geb.2009.05.004","volume":"71","author":"S Chien","year":"2011","unstructured":"Chien, S., Sinclair, A.: Convergence to approximate nash equilibria in congestion games. Games Econom. Behav. 71(2), 315\u2013327 (2011)","journal-title":"Games Econom. Behav."},{"key":"9_CR19","doi-asserted-by":"crossref","unstructured":"Caragiannis, I., Fanelli, A., Gravi, N., Skopalik, A.: Efficient computation of approximate pure nash equilibria in congestion games. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 532\u2013541 (2011)","DOI":"10.1109\/FOCS.2011.50"},{"issue":"1","key":"9_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/2614687","volume":"3","author":"I Caragiannis","year":"2015","unstructured":"Caragiannis, I., Fanelli, A., Gravin, N., Skopalik, A.: Approximate pure nash equilibria in weighted congestion games: existence, efficient computation, and structure. ACM Trans. Econ. Comput. (TEAC) 3(1), 1\u201332 (2015)","journal-title":"ACM Trans. Econ. Comput. (TEAC)"},{"key":"9_CR21","doi-asserted-by":"crossref","unstructured":"Feldotto, M., Gairing, M., Kotsialou, G., Skopalik, A.: Computing approximate pure nash equilibria in shapley value weighted congestion games. In: Proceedings of the 13th International Conference on Web and Internet Economics (WINE), pp. 191\u2013204 (2017)","DOI":"10.1007\/978-3-319-71924-5_14"},{"key":"9_CR22","unstructured":"Ren, C., Wu, Z., Xu, D., Yang, X.: $$\\frac{\\rho }{1-\\epsilon }$$-approximate pure nash equilibria algorithms for weighted congestion games and their runtimes, arXiv preprint arXiv:2208.11309"},{"issue":"1","key":"9_CR23","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/BF01386390","volume":"1","author":"EW Dijkstra","year":"1959","unstructured":"Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269\u2013271 (1959)","journal-title":"Numer. Math."}],"container-title":["Lecture Notes in Computer Science","Combinatorial Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-981-96-4448-3_9","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,3]],"date-time":"2025-05-03T05:50:54Z","timestamp":1746251454000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-981-96-4448-3_9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"ISBN":["9789819644476","9789819644483"],"references-count":23,"URL":"https:\/\/doi.org\/10.1007\/978-981-96-4448-3_9","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025]]},"assertion":[{"value":"4 May 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Disclosure of Interests"}},{"value":"COCOA","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Combinatorial Optimization and Applications","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Beijing","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"China","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2024","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"6 December 2024","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"8 December 2024","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"17","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"cocoa2024","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"http:\/\/www.cocoa2024.com\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}