Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints

We study the problem of allocating indivisible items to agents with additive valuations, under the additional constraint that bundles must be connected in an underlying item graph. Previous work has considered the existence and complexity of fair allocations. We study the problem of finding an allocation that is Pareto-optimal. While it is easy to find an efficient allocation when the underlying graph is a path or a star, the problem is NP-hard for many other graph topologies, even for trees of bounded pathwidth or of maximum degree 3. We show that on a path, there are instances where no Pareto-optimal allocation satisfies envy-freeness up to one good, and that it is NP-hard to decide whether such an allocation exists, even for binary valuations. We also show that, for a path, it is NP-hard to find a Pareto-optimal allocation that satisfies maximin share, but show that a moving-knife algorithm can find such an allocation when agents have binary valuations that have a non-nested interval structure.

[1]  Yingqian Zhang,et al.  On the Complexity of Efficiency and Envy-Freeness in Fair Division of Indivisible Goods with Additive Preferences , 2009, ADT.

[2]  Yonatan Aumann,et al.  The Efficiency of Fair Division with Connected Pieces , 2010, WINE.

[3]  Ariel D. Procaccia,et al.  Fairly Allocating Many Goods with Few Queries , 2018, AAAI.

[4]  Vincent Conitzer,et al.  Combinatorial Auctions with Structured Item Graphs , 2004, AAAI.

[5]  W. Stromquist.  How to Cut a Cake Fairly , 1980 .

[6]  Jörg Rothe,et al.  Cake-Cutting: Fair Division of Divisible Goods , 2016, Economics and Computation.

[7]  Rohit Vaish,et al.  Finding Fair and Efficient Allocations , 2017, EC.

[8]  Miroslaw Truszczynski,et al.  Maximin Share Allocations on Cycles , 2018, IJCAI.

[9]  Ariel D. Procaccia,et al.  The Unreasonable Fairness of Maximum Nash Welfare , 2016, EC.

[10]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[11]  Edith Elkind,et al.  Fair Division of a Graph , 2017, IJCAI.

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[13]  Kathryn L. Nyman,et al.  Discrete Envy-free Division of Necklaces and Maps , 2015, 1510.02132.

[14]  Laurent Gourvès,et al.  Object Allocation via Swaps along a Social Network , 2017, IJCAI.

[15]  Felix Brandt,et al.  Pareto optimality in coalition formation , 2011, Games Econ. Behav..

[16]  Eric Budish,et al.  The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes , 2010, Journal of Political Economy.

[17]  Vittorio Bilò,et al.  Almost Envy-Free Allocations with Connected Bundles , 2018, ITCS.

[18]  Jérôme Lang,et al.  Efficiency and envy-freeness in fair division of indivisible goods: logical representation and complexity , 2005, IJCAI.

[19]  Jon M. Kleinberg,et al.  Fair Division via Social Comparison , 2016, AAMAS.

[20]  Klaudia Frankfurter.  Computers And Intractability A Guide To The Theory Of Np Completeness , 2016 .

[21]  Rolf Niedermeier,et al.  Envy-Free Allocations Respecting Social Networks , 2018, AAMAS.

[22]  Jérôme Monnot,et al.  Optimal Reallocation under Additive and Ordinal Preferences , 2016, AAMAS.

[23]  Ning Chen,et al.  Optimal Proportional Cake Cutting with Connected Pieces , 2012, AAAI.

[24]  Avinatan Hassidim,et al.  Computing socially-efficient cake divisions , 2012, AAMAS.