{"type":"ScholarlyArticle","title":"The Complexity of Envy-Free Graph Cutting","authors":[{"name":"Argyrios Deligkas","slug":"argyrios-deligkas","given":"Argyrios","family":"Deligkas","affiliation":"Royal Holloway , University of London , UK"},{"name":"Eduard Eiben","slug":"eduard-eiben","given":"Eduard","family":"Eiben","affiliation":"Royal Holloway , University of London , UK"},{"name":"Robert Ganian","slug":"robert-ganian","given":"Robert","family":"Ganian","affiliation":"TU Wien , Austria"},{"name":"Thekla Hamm","slug":"thekla-hamm","given":"Thekla","family":"Hamm","affiliation":"TU Wien , Austria"},{"name":"Sebastian Ordyniak","slug":"sebastian-ordyniak","given":"Sebastian","family":"Ordyniak","affiliation":"University of Leeds , UK"}],"abstract":"We consider the problem of fairly dividing a set of heterogeneous divisible resources among agents with different preferences. We focus on the setting where the resources correspond to the edges of a connected graph, every agent must be assigned a connected piece of this graph, and the fairness notion considered is the classical envy freeness. The problem is NP-complete, and we analyze its complexity with respect to two natural complexity measures: the number of agents and the number of edges in the graph. While the problem remains NP-hard even for instances with 2 agents, we provide a dichotomy characterizing the complexity of the problem when the number of agents is constant based on structural properties of the graph. For the latter case, we design a polynomial-time algorithm when the graph has a constant number of edges.","doi":"10.24963/ijcai.2022/34","url":"https://ijcai.org/proceedings/2022/34","pdf":"https://ijcai.org/proceedings/2022/0034.pdf","markdown":"https://ijcai.org/proceedings/2022/34.md","bibtex":"https://ijcai.org/proceedings/2022/bibtex/34","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":34,"pages":{"first":237,"last":243},"keywords":["Agent-based and Multi-agent Systems: Resource Allocation","Agent-based and Multi-agent Systems: Computational Social Choice"],"videos":[{"label":"Video #1","url":"https://slideslive.com/38986725","length_sec":85,"provider":"slideslive"},{"label":"Video #2","url":"https://slideslive.com/38984868","length_sec":353,"provider":"slideslive"}],"references":[{"position":1,"raw":"[Abebe et al., 2017] Rediet Abebe, Jon Kleinberg, and David C. Parkes. Fair division via social comparison. In AAMAS, page 281-289, 2017.","title":"Fair division via social comparison","authors":"Abebe","year":"2017","venue":"AAMAS","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Aumann and Dombb, 2015] Yonatan Aumann and Yair Dombb. The efficiency of fair division with connected pieces. ACM TEAC, 3(4):1-16, 2015.","title":"The efficiency of fair division with connected pieces","authors":"Aumann, Dombb, Yonatan Aumann, Yair Dombb","year":"2015","venue":"ACM TEAC","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Aziz et al., 2018] Haris Aziz, Sylvain Bouveret, Ioannis Caragiannis, Ira Giagkousi, and Jérôme Lang. Knowledge, fairness, and social constraints. In AAAI, number 1, 2018.","title":"Knowledge, fairness, and social constraints","authors":"Haris Aziz, Sylvain Bouveret, Ioannis Caragiannis, Ira Giagkousi, Jérôme Lang","year":"2018","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Balkanski et al., 2014] E. Balkanski, S. Brânzei, D. Kurokawa, and A. Procaccia. Simultaneous cake cutting. In AAAI, pages 566-572, 2014.","title":"Simultaneous cake cutting","authors":"E Balkanski, S Brânzei, D Kurokawa, A Procaccia","year":"2014","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Basu et al., 2006] Saugata Basu, Richard Pollack, and Marie-Franc ¸oise Roy. Algorithms in Real Algebraic Ge- ometry. Springer-Verlag, 2006.","title":"Algorithms in Real Algebraic Geometry","authors":"Saugata Basu, Richard Pollack, Marie-Franc ¸oise Roy","year":"2006","venue":null,"doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Bei and Suksompong, 2021] Xiaohui Bei and Warut Suk- sompong. Dividing a graphical cake. In AAAI, pages 5159-5166, 2021.","title":"Dividing a graphical cake","authors":"Xiaohui Bei, Warut Suksompong","year":"2021","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Bei et al., 2017] Xiaohui Bei, Youming Qiao, and Shengyu Zhang. Networked fairness in cake cutting. In IJCAI, pages 3632-3638, 2017.","title":"Networked fairness in cake cutting","authors":"Xiaohui Bei, Youming Qiao, Shengyu Zhang","year":"2017","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2017/508"},{"position":8,"raw":"[Bei et al., 2019] Xiaohui Bei, Ayumi Igarashi, Xinhang Lu, and Warut Suksompong. Connected fair allocation of in- divisible goods. arXiv preprint arXiv:1908.05433, 2019.","title":"Connected fair allocation of indivisible goods","authors":"Xiaohui Bei, Ayumi Igarashi, Xinhang Lu, Warut Suksompong","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Bei et al., 2020] Xiaohui Bei, Xiaoming Sun, Hao Wu, Jialin Zhang, Zhijie Zhang, and Wei Zi. Cake cutting on graphs: A discrete and bounded proportional protocol. In SODA, pages 2114-2123, 2020.","title":"Cake cutting on graphs: A discrete and bounded proportional protocol","authors":"Xiaohui Bei, Xiaoming Sun, Hao Wu, Jialin Zhang, Zhijie Zhang, Wei Zi","year":"2020","venue":"SODA","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Bei et al., 2021] Xiaohui Bei, Ayumi Igarashi, Xinhang Lu, and Warut Suksompong. The price of connectivity in fair division. In AAAI, pages 5151-5158, 2021.","title":"The price of connectivity in fair division","authors":"Xiaohui Bei, Ayumi Igarashi, Xinhang Lu, Warut Suksompong","year":"2021","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Bilò et al., 2021] Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, and William S Zwicker. Almost envy-free allocations with connected bundles. Games and Economic Behavior, 2021.","title":"Almost envy-free allocations with connected bundles","authors":"Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, William S Zwicker","year":"2021","venue":"Games and Economic Behavior","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Bouveret et al., 2017] Sylvain Bouveret, Katarína Cechlárová, Edith Elkind, Ayumi Igarashi, and Do- minik Peters. Fair division of a graph. In IJCAI, pages 135-141, 2017.","title":"Fair division of a graph","authors":"Sylvain Bouveret, Katarína Cechlárová, Edith Elkind, Ayumi Igarashi, Dominik Peters","year":"2017","venue":"IJCAI","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Brams and Taylor, 1996] Steven J Brams and Alan D Tay- lor. Fair Division: From cake-cutting to dispute resolution. Cambridge University Press, 1996.","title":"Fair Division: From cake-cutting to dispute resolution","authors":"J Steven, Alan D Brams, Taylor","year":"1996","venue":null,"doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Brânzei and Miltersen, 2015] Simina Brânzei and Peter Bro Miltersen. A dictatorship theorem for cake cutting. In IJCAI, pages 482-488, 2015.","title":"A dictatorship theorem for cake cutting","authors":"Simina Brânzei, Peter Bro Miltersen","year":"2015","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2015/74"},{"position":15,"raw":"[Caragiannis et al., 2011] Ioannis Caragiannis, John K. Lai, and Ariel D. Procaccia. Towards more expressive cake cutting. In IJCAI, pages 127-132, 2011.","title":"Towards more expressive cake cutting","authors":"Ioannis Caragiannis, John K Lai, Ariel D Procaccia","year":"2011","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2011/33"},{"position":16,"raw":"[Chevaleyre et al., 2017] Yann Chevaleyre, Ulle Endriss, and Nicolas Maudet. Distributed fair allocation of indi- visible goods. Artificial Intelligence, 242:1-22, 2017.","title":"Distributed fair allocation of indivisible goods","authors":"Yann Chevaleyre, Ulle Endriss, Nicolas Maudet","year":"2017","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Cohen et al., 2019] Michael B. Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. In STOC, pages 938-942, 2019.","title":"Solving linear programs in the current matrix multiplication time","authors":"B Michael, Yin Cohen, Zhao Tat Lee, Song","year":"2019","venue":"STOC","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Deligkas et al., 2021] Argyrios Deligkas, Eduard Eiben, Robert Ganian, Thekla Hamm, and Sebastian Ordyniak. The parameterized complexity of connected fair division. In IJCAI, pages 139-145, 2021.","title":"The parameterized complexity of connected fair division","authors":"Deligkas","year":"2021","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2021/20"},{"position":19,"raw":"[Dubins and Spanier, 1961] Lester E Dubins and Edwin H Spanier. How to cut a cake fairly. The American Math- ematical Monthly, 68(1P1):1-17, 1961.","title":"How to cut a cake fairly","authors":"E Lester, Edwin H Dubins, Spanier","year":"1961","venue":"The American Mathematical Monthly","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Eiben et al., 2020] Eduard Eiben, Robert Ganian, Thekla Hamm, and Sebastian Ordyniak. Parameterized complex- ity of envy-free resource allocation in social networks. In AAAI, pages 7135-7142, 2020.","title":"Parameterized complexity of envy-free resource allocation in social networks","authors":"Eiben","year":"2020","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Elkind et al., 2021a] Edith Elkind, Erel Segal-Halevi, and Warut Suksompong. Graphical cake cutting via maximin share. In IJCAI, pages 161-167, 2021.","title":"Graphical cake cutting via maximin share","authors":"Elkind","year":"2021","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2021/23"},{"position":22,"raw":"[Elkind et al., 2021b] Edith Elkind, Erel Segal-Halevi, and Warut Suksompong. Keep your distance: Land division with separation. In IJCAI, pages 168-174, 2021.","title":"Keep your distance: Land division with separation","authors":"Elkind","year":"2021","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2021/24"},{"position":23,"raw":"[Elkind et al., 2021c] Edith Elkind, Erel Segal-Halevi, and Warut Suksompong. Mind the gap: Cake cutting with sep- aration. In AAAI, pages 5330-5338, 2021.","title":"Mind the gap: Cake cutting with separation","authors":"Elkind","year":"2021","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Igarashi and Peters, 2019] Ayumi Igarashi and Dominik Pe- ters. Pareto-optimal allocation of indivisible goods with connectivity constraints. In AAAI, pages 2045-2052, 2019.","title":"Ayumi Igarashi and Dominik Peters. Pareto-optimal allocation of indivisible goods with connectivity constraints","authors":"Peters Igarashi","year":"2019","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Marenco and Tetzlaff, 2014] Javier Marenco and Tomás Tetzlaff. Envy-free division of discrete cakes. Discrete Applied Mathematics, 164:527-531, 2014.","title":"Envy-free division of discrete cakes","authors":"Tetzlaff Marenco, Javier Marenco, Tomás Tetzlaff","year":"2014","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Menon and Larson, 2017] Vijay Menon and Kate Larson. Deterministic, strategyproof, and fair cake cutting. In IJ- CAI, pages 352-358, 2017.","title":"Deterministic, strategyproof, and fair cake cutting","authors":"Vijay Menon, Kate Larson","year":"2017","venue":"IJ-CAI","doi":null,"url":null,"internal_url":"/proceedings/2017/50"},{"position":27,"raw":"[Moulin, 2004] Hervé Moulin. Fair division and collective welfare. MIT press, 2004.","title":"Fair division and collective welfare","authors":"Hervé Moulin","year":"2004","venue":null,"doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Procaccia, 2013] Ariel D Procaccia. Cake cutting: Not just child's play. Commun. of the ACM, 56(7):78-87, 2013.","title":"Cake cutting: Not just child's play","authors":"D Ariel, Procaccia","year":"2013","venue":"Commun. of the ACM","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Robertson and Webb, 1998] Jack Robertson and William Webb. Cake-cutting algorithms: Be fair if you can. CRC Press, 1998.","title":"Cake-cutting algorithms: Be fair if you can","authors":"Webb ; Robertson, Jack Robertson, William Webb","year":"1998","venue":null,"doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Segal-Halevi et al., 2016] Erel Segal-Halevi, Avinatan Has- sidim, and Yonatan Aumann. Waste makes haste: bounded time algorithms for envy-free cake cutting with free dis- posal. Transactions on Algorithms, 13(1):1-32, 2016.","title":"Waste makes haste: bounded time algorithms for envy-free cake cutting with free disposal","authors":"Erel Segal-Halevi, Avinatan Hassidim, Yonatan Aumann","year":"2016","venue":"Transactions on Algorithms","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Simonov et al., 2019] Kirill Simonov, Fedor V. Fomin, Petr A. Golovach, and Fahad Panolan. Refined complexity of PCA with outliers. In ICML, pages 5818-5826, 2019.","title":"Refined complexity of PCA with outliers","authors":"Kirill Simonov, V Fedor, Petr A Fomin, Fahad Golovach, Panolan","year":"2019","venue":"ICML","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Stromquist, 1980] Walter Stromquist. How to cut a cake fairly. Am. Math. Mon., 87(8):640-644, 1980.","title":"How to cut a cake fairly","authors":"Walter Stromquist","year":"1980","venue":"Am. Math. Mon","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}