Computing an Approximately Optimal Agreeable Set of Items

Abstract We study the problem of assigning a small subset of indivisible items to a group of agents so that the subset is agreeable to all agents, meaning that all agents value the subset as least as much as its complement. For an arbitrary number of agents and items, we derive a tight worst-case bound on the number of items that may need to be included in such a set. We then present polynomial-time algorithms that find an agreeable set whose size matches the worst-case bound when there are two or three agents. We also show that finding small agreeable sets is possible even when we only have access to the agents' preferences on single items. Furthermore, we investigate the problem of efficiently computing an agreeable set whose size approximates the size of the smallest agreeable set for any given instance. We consider two well-known models for representing the preferences of the agents—the value oracle model and additive utilities—and establish tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time in each of these models.

[1]  P. Levy.  Théorie de l'addition des variables aléatoires , 1955 .

[2]  U. Feige,et al.  Maximizing Non-monotone Submodular Functions , 2011 .

[3]  Pasin Manurangsi,et al.  Computing an Approximately Optimal Agreeable Set of Items , 2017, IJCAI.

[4]  Aleksandar Nikolov,et al.  Beck's Three Permutations Conjecture: A Counterexample and Some Consequences , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[5]  Erel Segal-Halevi,et al.  Proportional Cake-Cutting among Families , 2015 .

[6]  Ariel D. Procaccia,et al.  When Can the Maximin Share Guarantee Be Guaranteed? , 2016, AAAI.

[7]  Pasin Manurangsi,et al.  Asymptotic existence of fair divisions for groups , 2017, Math. Soc. Sci..

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

[9]  Géza Bohus,et al.  On the Discrepancy of 3 Permutations , 1990, Random Struct. Algorithms.

[10]  Vincent Conitzer,et al.  A double oracle algorithm for zero-sum security games on graphs , 2011, AAMAS.

[11]  Joseph Y. Halpern,et al.  Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence , 2014, AAAI 2014.

[12]  Ariel D. Procaccia,et al.  No agent left behind: dynamic fair division of multiple resources , 2013, AAMAS.

[13]  Siddharth Barman,et al.  Fair Division Under Cardinality Constraints , 2018, IJCAI.

[14]  Lirong Xia,et al.  A Dichotomy Theorem on the Existence of Efficient or Neutral Sequential Voting Correspondences , 2009, IJCAI.

[15]  Ariel D. Procaccia,et al.  On Maxsum Fair Cake Divisions , 2012, AAAI.

[16]  P. Pattanaik,et al.  Social choice and welfare , 1983 .

[17]  Francis Edward Su,et al.  Consensus-halving via theorems of Borsuk-Ulam and Tucker , 2003, Math. Soc. Sci..

[18]  Evangelos Markakis,et al.  Approximation Algorithms for Computing Maximin Share Allocations , 2015, ICALP.

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

[20]  Eth Zentrum,et al.  A Combinatorial Proof of Kneser's Conjecture , 2022 .

[21]  Ulrich Endriss,et al.  Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods , 2010, ECAI.

[22]  Nick Bostrom,et al.  Thinking Inside the Box: Controlling and Using an Oracle AI , 2012, Minds and Machines.

[23]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[24]  Elchanan Mossel,et al.  On approximately fair allocations of indivisible goods , 2004, EC '04.

[25]  Joan Feigenbaum,et al.  Proceedings of the 5th ACM conference on Electronic commerce , 2004 .

[26]  Tim Roughgarden,et al.  Welfare guarantees for combinatorial auctions with item bidding , 2011, SODA '11.

[27]  David Steurer,et al.  Analytical approach to parallel repetition , 2013, STOC.

[28]  Ariel D. Procaccia,et al.  Optimal Envy-Free Cake Cutting , 2011, AAAI.

[29]  E. Weisstein.  Kneser's Conjecture , 2002 .

[30]  Haris Aziz,et al.  Justified representation in approval-based committee voting , 2014, Social Choice and Welfare.

[31]  Uriel Feige,et al.  On maximizing welfare when utility functions are subadditive , 2006, STOC '06.

[32]  Laurent Gourvès,et al.  Agreeable sets with matroidal constraints , 2018, J. Comb. Optim..

[33]  Toby Walsh,et al.  Fair assignment of indivisible objects under ordinal preferences , 2013, AAMAS.

[34]  Raphaël Clifford,et al.  ACM-SIAM Symposium on Discrete Algorithms , 2015, SODA 2015.

[35]  Vincent Conitzer,et al.  Introduction to Computational Social Choice , 2016, Handbook of Computational Social Choice.

[36]  SaberiAmin,et al.  Approximation Algorithms for Computing Maximin Share Allocations , 2017 .

[37]  Haris Aziz,et al.  A note on the undercut procedure , 2013, AAMAS.

[38]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[39]  Ariel D. Procaccia,et al.  The Computational Rise and Fall of Fairness , 2014, AAAI.

[40]  Ronald V. Book,et al.  Review: Michael R. Garey and David S. Johnson, Computers and intractability: A guide to the theory of $NP$-completeness , 1980 .

[41]  Yann Chevaleyre,et al.  Issues in Multiagent Resource Allocation , 2006, Informatica.

[42]  Uriel Feige.  On Maximizing Welfare When Utility Functions Are Subadditive , 2009, SIAM J. Comput..

[43]  H. Varian.  Equity, Envy and Efficiency , 1974 .

[44]  Jstor,et al.  Proceedings of the American Mathematical Society , 1950 .

[45]  Noga Alon,et al.  Algorithmic construction of sets for k-restrictions , 2006, TALG.

[46]  Sylvain Bouveret,et al.  Characterizing conflicts in fair division of indivisible goods using a scale of criteria , 2016, Autonomous Agents and Multi-Agent Systems.

[47]  Ramaswamy Chandrasekaran,et al.  A Class of Sequential Games , 1971, Oper. Res..

[48]  D. Foley.  Resource allocation and the public sector , 1967 .

[49]  N. Alon,et al.  The Borsuk-Ulam theorem and bisection of necklaces , 1986 .

[50]  Steven J. Brams,et al.  The undercut procedure: an algorithm for the envy-free division of indivisible items , 2009, Soc. Choice Welf..

[51]  Kirk Pruhs,et al.  Divorcing Made Easy , 2012, FUN.

[52]  Dana Moshkovitz,et al.  The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover , 2012, Theory Comput..

[53]  Warut Suksompong,et al.  Democratic Fair Allocation of Indivisible Goods , 2017, IJCAI.

[54]  Erel Segal-Halevi,et al.  Envy-Free Cake-Cutting among Families , 2016, ArXiv.

[55]  Stavros G. Kolliopoulos,et al.  Approximation Algorithms for Covering/Packing Integer Programs , 2002, cs/0205030.

[56]  Nils J. Nilsson,et al.  Artificial Intelligence , 1974, IFIP Congress.

[57]  Joshua Evan Greene,et al.  A New Short Proof of Kneser's Conjecture , 2002, Am. Math. Mon..

[58]  Ariel D. Procaccia,et al.  The Unreasonable Fairness of Maximum Nash Welfare , 2019, ACM Trans. Economics and Comput..

[59]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[60]  Warut Suksompong.  Assigning a Small Agreeable Set of Indivisible Items to Multiple Players , 2016, IJCAI.

[61]  Warut Suksompong,et al.  Approximate maximin shares for groups of agents , 2017, Math. Soc. Sci..

[62]  Jirí Matousek,et al.  A Combinatorial Proof of Kneser’s Conjecture* , 2004, Comb..

[63]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[64]  Piotr Faliszewski,et al.  Finding a collective set of items: From proportional multirepresentation to group recommendation , 2014, Artif. Intell..

[65]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[66]  K. Arrow,et al.  Handbook of Social Choice and Welfare , 2011 .

[67]  Jérôme Lang,et al.  Logical Preference Representation and Combinatorial Vote , 2004, Annals of Mathematics and Artificial Intelligence.

[68]  Imre Bárány,et al.  A Short Proof of Kneser's Conjecture , 1978, J. Comb. Theory, Ser. A.

[69]  Peter C. Fishburn,et al.  Fair division of indivisible items between two people with identical preferences: Envy-freeness, Pareto-optimality, and equity , 2000, Soc. Choice Welf..

[70]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[71]  W. Bossert,et al.  Ranking Sets of Objects , 2001 .

[72]  László Lovász,et al.  Kneser's Conjecture, Chromatic Number, and Homotopy , 1978, J. Comb. Theory A.

[73]  Avrim Blum,et al.  Planning in the Presence of Cost Functions Controlled by an Adversary , 2003, ICML.