{"type":"ScholarlyArticle","title":"Voting in Two-Crossing Elections","authors":[{"name":"Andrei Constantinescu","slug":"andrei-constantinescu","given":"Andrei","family":"Constantinescu","affiliation":"ETH Zürich"},{"name":"Roger Wattenhofer","slug":"roger-wattenhofer","given":"Roger","family":"Wattenhofer","affiliation":"ETH Zürich"}],"abstract":"We introduce two-crossing elections as a generalization of single-crossing elections, showing a number of new results. First, we show that two-crossing elections can be recognized in polynomial time, by reduction to the well-studied consecutive ones problem. Single-crossing elections exhibit a transitive majority relation, from which many important results follow. On the other hand, we show that the classical Debord-McGarvey theorem can still be proven two-crossing, implying that any weighted majority tournament is inducible by a two-crossing election. This shows that many voting rules are NP-hard under two-crossing elections, including Kemeny and Slater. This is in contrast to the single-crossing case and outlines an important complexity boundary between single- and two-crossing. Subsequently, we show that for two-crossing elections the Young scores of all candidates can be computed in polynomial time, by formulating a totally unimodular linear program. Finally, we consider the Chamberlin-Courant rule with arbitrary disutilities and show that a winning committee can be computed in polynomial time, using an approach based on dynamic programming.","doi":"10.24963/ijcai.2022/30","url":"https://ijcai.org/proceedings/2022/30","pdf":"https://ijcai.org/proceedings/2022/0030.pdf","markdown":"https://ijcai.org/proceedings/2022/30.md","bibtex":"https://ijcai.org/proceedings/2022/bibtex/30","volume":{"slug":"2022","title":"Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2022","url":"https://ijcai.org/proceedings/2022/","year":2022,"location":"Vienna","dates":"23-29 July 2022","editors":"Luc De Raedt","isbn":"978-1-956792-00-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2022"},"section":"Agent-based and Multi-agent Systems","track":"Main Track","number":30,"pages":{"first":208,"last":214},"keywords":["Agent-based and Multi-agent Systems: Computational Social Choice"],"videos":[{"label":"Video #1","url":"https://slideslive.com/38987032","length_sec":81,"provider":"slideslive"},{"label":"Video #2","url":"https://slideslive.com/38985175","length_sec":711,"provider":"slideslive"}],"references":[{"position":1,"raw":"[Bartholdi et al., 1989] J. Bartholdi, C. A. Tovey, and M. A. Trick. Voting schemes for which it can be difficult to tell who won the election. Social Choice and Welfare, 6(2):157-165, 1989.","title":"Voting schemes for which it can be difficult to tell who won the election","authors":"J Bartholdi, C A Tovey, M A Trick","year":"1989","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Booth and Lueker, 1976] Kellogg S. Booth and George S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. Journal of Computer and System Sciences, 13(3):335-379, 1976.","title":"Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms","authors":"S Kellogg, George S Booth, Lueker","year":"1976","venue":"Journal of Computer and System Sciences","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Brandt et al., 2015] Felix Brandt, Markus Brill, Edith Hemaspaan- dra, and Lane A. Hemaspaandra. Bypassing combinatorial protections: polynomial-time algorithms for single-peaked elec- torates. J. Artif. Int. Res., 53(1):439-496, may 2015.","title":"Bypassing combinatorial protections: polynomial-time algorithms for single-peaked electorates","authors":"Felix Brandt, Markus Brill, Edith Hemaspaandra, Lane A Hemaspaandra","year":"2015","venue":"J. Artif. Int. Res","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Brandt et al., 2016] Felix Brandt, Markus Brill, and Paul Harren- stein. Tournament solutions. In Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia, editors, Handbook of Computational Social Choice, chapter 3. Cam- bridge University Press, USA, 1st edition, 2016.","title":"Tournament solutions","authors":"Felix Brandt, Markus Brill, Paul Harrenstein","year":"2016","venue":"Handbook of Computational Social Choice, chapter 3","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Bredereck et al., 2013] Robert Bredereck, Jiehua Chen, and Ger- hard J. Woeginger. A characterization of the single-crossing do- main. Social Choice and Welfare, 41(4):989-998, 2013.","title":"A characterization of the single-crossing domain","authors":"Robert Bredereck, Jiehua Chen, Gerhard J Woeginger","year":"2013","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Caragiannis et al., 2016] Ioannis Caragiannis, Edith Hemaspaan- dra, and Lane A. Hemaspaandra. Dodgson's rule and young's rule. In Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia, editors, Handbook of Compu- tational Social Choice, chapter 5. Cambridge University Press, USA, 1st edition, 2016.","title":"Dodgson's rule and young's rule","authors":"Ioannis Caragiannis, Edith Hemaspaandra, Lane A Hemaspaandra","year":"2016","venue":"Handbook of Computational Social Choice","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Chamberlin and Courant, 1983] John Chamberlin and Paul Courant. Representative deliberations and representative deci- sions: proportional representation and the borda rule. American Political Science Review, 77(3):718-733, 1983.","title":"John Chamberlin and Paul Courant. Representative deliberations and representative decisions: proportional representation and the borda rule","authors":"Courant Chamberlin","year":"1983","venue":"American Political Science Review","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Chen et al., 2021] Chia-Hui Chen, Junichiro Ishida, and Wing Suen. Signaling under Double-Crossing Preferences. ISER Dis- cussion Paper 1103rr, Institute of Social and Economic Research, Osaka University, Oct 2021.","title":"Signaling under Double-Crossing Preferences","authors":"Chia-Hui Chen, Junichiro Ishida, Wing Suen","year":"2021","venue":null,"doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Constantinescu and Elkind, 2021] Andrei Costin Constantinescu and Edith Elkind. Proportional representation under single- crossing preferences revisited. Proceedings of the AAAI Con- ference on Artificial Intelligence, 35(6):5286-5293, May 2021.","title":"Proportional representation under singlecrossing preferences revisited","authors":"Elkind Constantinescu, Andrei Costin Constantinescu, Edith Elkind","year":"2021","venue":"Proceedings of the AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Constantinescu and Wattenhofer, 2022] Andrei Costin Constanti- nescu and Roger Wattenhofer. Voting in two-crossing elections. arXiv preprint, arXiv:2205.00474, 2022.","title":"Voting in two-crossing elections","authors":"Andrei, Costin Constantinescu, Roger Wattenhofer","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Elkind et al., 2012] Edith Elkind, Piotr Faliszewski, and Arkadii Slinko. Clone structures in voters' preferences. Proceedings of the ACM Conference on Electronic Commerce, pages 496-513, 2012.","title":"Clone structures in voters' preferences","authors":"Elkind","year":"2012","venue":"Proceedings of the ACM Conference on Electronic Commerce","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Faliszewski et al., 2017] Piotr Faliszewski, Piotr Skowron, Arkadii Slinko, and Nimrod Talmon. Multiwinner voting: a new challenge for social choice theory. In Ulle Endriss, editor, Trends in Computational Social Choice, chapter 2, pages 27-47. AI Access, 2017.","title":"Multiwinner voting: a new challenge for social choice theory","authors":"Piotr Faliszewski, Piotr Skowron, Arkadii Slinko, Nimrod Talmon","year":"2017","venue":"Trends in Computational Social Choice","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Fischer et al., 2016] Felix Fischer, Olivier Hudry, and Rolf Nieder- meier. Weighted tournament solutions. In Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia, edi- tors, Handbook of Computational Social Choice, chapter 4. Cam- bridge University Press, USA, 1st edition, 2016.","title":"Weighted tournament solutions","authors":"Felix Fischer, Olivier Hudry, Rolf Niedermeier","year":"2016","venue":"Handbook of Computational Social Choice, chapter 4","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Fitzsimmons and Hemaspaandra, 2020] Zack Fitzsimmons and Edith Hemaspaandra. Election score can be harder than winner. In ECAI 2020, 2020.","title":"Zack Fitzsimmons and Edith Hemaspaandra. Election score can be harder than winner","authors":"Hemaspaandra Fitzsimmons","year":"2020","venue":"ECAI 2020","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Fulkerson and Gross, 1965] Delbert Ray Fulkerson and Oliver Al- fred Gross. Incidence matrices and interval graphs. Pacific Jour- nal of Mathematics, 15:835-855, 1965.","title":"Delbert Ray Fulkerson and Oliver Alfred Gross. Incidence matrices and interval graphs","authors":"Gross Fulkerson","year":"1965","venue":"Pacific Journal of Mathematics","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Goldberg et al., 1995] Paul Goldberg, Martin Golumbic, Haim Kaplan, and Ron Shamir. Four strikes against physical mapping of dna. Journal of computational biology: a journal of computa- tional molecular cell biology, 2:139-52, 03 1995.","title":"Four strikes against physical mapping of dna","authors":"Paul Goldberg, Martin Golumbic, Haim Kaplan, Ron Shamir","year":"1995","venue":"Journal of computational biology: a journal of computational molecular cell biology","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Hemaspaandra et al., 1997] Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. Exact analysis of dodgson elections: lewis carroll's 1876 voting system is complete for parallel access to np. J. ACM, 44(6):806-825, Nov 1997.","title":"Exact analysis of dodgson elections: lewis carroll's 1876 voting system is complete for parallel access to np","authors":"Hemaspaandra","year":"1997","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Hemaspaandra et al., 2005] Edith Hemaspaandra, Holger Spakowski, and Jörg Vogel. The complexity of kemeny elections. Theoretical Computer Science, 349:382-391, 12 2005.","title":"The complexity of kemeny elections","authors":"Hemaspaandra","year":"2005","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Hoffman and Kruskal, 2016] A. J. Hoffman and J. B. Kruskal. In- tegral boundary points of convex polyhedra. In Linear Inequali- ties and Related Systems. (AM-38), Volume 38, chapter 13, pages 223-246. Princeton University Press, 2016.","title":"Integral boundary points of convex polyhedra","authors":"A J Hoffman, J B","year":"2016","venue":"Linear Inequalities and Related Systems","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Karmarkar, 1984] Narendra Karmarkar. A new polynomial-time algorithm for linear programming. In Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, STOC'84, pages 302-311, New York, NY, USA, 1984. Associa- tion for Computing Machinery.","title":"A new polynomial-time algorithm for linear programming","authors":"Narendra Karmarkar","year":"1984","venue":"Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, STOC'84","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Lu and Boutilier, 2011] Tyler Lu and Craig Boutilier. Budgeted social choice: from consensus to personalized decision making. In Proceedings of IJCAI'11, pages 280-286, 01 2011.","title":"Budgeted social choice: from consensus to personalized decision making","authors":"Lu, Boutilier, Tyler Lu, Craig Boutilier","year":"2011","venue":"Proceedings of IJCAI'11","doi":null,"url":null,"internal_url":"/proceedings/2011/57"},{"position":22,"raw":"[Mirrlees, 1971] James Mirrlees. An exploration in the theory of optimum income taxation. The Review of Economic Studies, 38(2):175-208, 1971.","title":"An exploration in the theory of optimum income taxation","authors":"James Mirrlees","year":"1971","venue":"The Review of Economic Studies","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Misra et al., 2017] Neeldhara Misra, Chinmay Sonar, and P. R. Vaidyanathan. On the complexity of chamberlin-courant on al- most structured profiles. In Jörg Rothe, editor, Algorithmic Deci- sion Theory, pages 124-138, Cham, 2017. Springer International Publishing.","title":"On the complexity of chamberlin-courant on almost structured profiles","authors":"Misra","year":"2017","venue":"Algorithmic Decision Theory","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Peters and Lackner, 2020] Dominik Peters and Martin Lackner. Preferences single-peaked on a circle. Journal of Artificial In- telligence Research, 68:463-502, 06 2020.","title":"Preferences single-peaked on a circle","authors":"Lackner Peters, Dominik Peters, Martin Lackner","year":"2020","venue":"Journal of Artificial Intelligence Research","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Procaccia et al., 2008] Ariel Procaccia, Jeffrey Rosenschein, and Aviv Zohar. On the complexity of achieving proportional repre- sentation. Social Choice and Welfare, 30:353-362, 02 2008.","title":"On the complexity of achieving proportional representation","authors":"Procaccia","year":"2008","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Roberts, 1977] Kevin Roberts. Voting over income tax schedules. Journal of Public Economics, 8(3):329 -340, 1977.","title":"Voting over income tax schedules","authors":"Kevin Roberts","year":"1977","venue":"Journal of Public Economics","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Rothe et al., 2003] Jörg Rothe, Holger Spakowski, and Jörg Vo- gel. Exact complexity of the winner problem for young elections. Theory of Computing Systems, 36:375-386, 2003.","title":"Exact complexity of the winner problem for young elections","authors":"Rothe","year":"2003","venue":"Theory of Computing Systems","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Skowron et al., 2015] Piotr Skowron, Lan Yu, Piotr Faliszewski, and Edith Elkind. The complexity of fully proportional repre- sentation for single-crossing electorates. Theoretical Computer Science, 569:43-57, 2015.","title":"The complexity of fully proportional representation for single-crossing electorates","authors":"Piotr Skowron, Lan Yu, Piotr Faliszewski, Edith Elkind","year":"2015","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Tucker, 1971] Alan C. Tucker. Matrix characterizations of circular-arc graphs. Pacific Journal of Mathematics, 39:535-545, 1971.","title":"Matrix characterizations of circular-arc graphs","authors":"Alan C Tucker","year":"1971","venue":"Pacific Journal of Mathematics","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}