37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, December 11-15, 2017, Kanpur, India

Suppose that a group of voters wants to select k ≥ 1 alternatives from a given set, and each voter indicates which of the alternatives are acceptable to her: the alternatives could be conference submissions, applicants for a scholarship or locations for a fast food chain. In this setting it is natural to require that the winning set represents the voters fairly, in the sense that large groups of voters with similar preferences have at least some of their approved alternatives in the winning set. We describe several ways to formalize this idea, and show how to use it to classify voting rules; surprisingly, two voting rules proposed in the XIXth century turn out to play an important role in our analysis. 1998 ACM Subject Classification F.2 Analysis of algorithms and problem complexity

[1]  Dana S. Scott,et al.  Finite Automata and Their Decision Problems , 1959, IBM J. Res. Dev..

[2]  G. Dirac.  On rigid circuit graphs , 1961 .

[3]  Marvin Minsky,et al.  Perceptrons: An Introduction to Computational Geometry , 1969 .

[4]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[5]  Peter van Emde Boas.  Preserving order in a forest in less than logarithmic time , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[6]  P. Gärdenfors.  Match making: Assignments based on bilateral preferences , 1975 .

[7]  P. Hammer,et al.  Aggregation of inequalities in integer programming. , 1975 .

[8]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[9]  David S. Johnson,et al.  The Rectilinear Steiner Tree Problem is NP Complete , 1977, SIAM Journal of Applied Mathematics.

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

[11]  Martin Charles Golumbic,et al.  Trivially perfect graphs , 1978, Discret. Math..

[12]  Andrew Chi-Chih Yao,et al.  Should Tables Be Sorted? , 1981, JACM.

[13]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[14]  John M. Lewis,et al.  The Node-Deletion Problem for Hereditary Properties is NP-Complete , 1980, J. Comput. Syst. Sci..

[15]  Mihalis Yannakakis,et al.  Node-Deletion Problems on Bipartite Graphs , 1981, SIAM J. Comput..

[16]  M. Yannakakis.  Computing the Minimum Fill-in is NP^Complete , 1981 .

[17]  Alan A. Bertossi,et al.  Dominating Sets for Split and Bipartite Graphs , 1984, Inf. Process. Lett..

[18]  Martin Aigner,et al.  A game of cops and robbers , 1984, Discret. Appl. Math..

[19]  A. Roth.  Stability and Polarization of Interests in Job Matching , 1984 .

[20]  D. T. Lee,et al.  Computational complexity of art gallery problems , 1986, IEEE Trans. Inf. Theory.

[21]  Peter Frankl,et al.  Complexity classes in communication complexity theory , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[22]  Mihalis Yannakakis,et al.  The Maximum k-Colorable Subgraph Problem for Chordal Graphs , 1987, Inf. Process. Lett..

[23]  David E. Muller,et al.  Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[24]  Alok Aggarwal,et al.  The input/output complexity of sorting and related problems , 1988, CACM.

[25]  Charles Blair,et al.  The Lattice Structure of the Set of Stable Matchings with Multiple Partners , 1988, Math. Oper. Res..

[26]  Robert W. Irving,et al.  The Stable marriage problem - structure and algorithms , 1989, Foundations of computing series.

[27]  Michael E. Saks,et al.  The cell probe complexity of dynamic data structures , 1989, STOC '89.

[28]  Jean-Claude Latombe,et al.  Robot motion planning , 1991, The Kluwer international series in engineering and computer science.

[29]  Zhi-Li Zhang.  Complexity of Symmetric Functions in Perceptron-Like Models , 1992 .

[30]  Noam Nisan,et al.  On the degree of boolean functions as real polynomials , 1992, STOC '92.

[31]  Ramamohan Paturi,et al.  On the degree of polynomials that approximate symmetric Boolean functions (preliminary version) , 1992, STOC '92.

[32]  Jens Gustedt,et al.  On the Pathwidth of Chordal Graphs , 1993, Discret. Appl. Math..

[33]  Mario Szegedy.  Functions with Bounded Symmetric Communication Complexity, Programs over Commutative Monoids, and ACC , 1993, J. Comput. Syst. Sci..

[34]  Carsten Lund,et al.  The Approximation of Maximum Subgraph Problems , 1993, ICALP.

[35]  Pierre Wolper,et al.  Reasoning About Infinite Computations , 1994, Inf. Comput..

[36]  Steven Skiena,et al.  Complexity aspects of visibility graphs , 1995, Int. J. Comput. Geom. Appl..

[37]  Wolfgang Thomas,et al.  On the Synthesis of Strategies in Infinite Games , 1995, STACS.

[38]  Leizhen Cai,et al.  Fixed-Parameter Tractability of Graph Modification Problems for Hereditary Properties , 1996, Inf. Process. Lett..

[39]  Orna Kupferman,et al.  Relating word and tree automata , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[40]  Haiko Müller,et al.  Hamiltonian circuits in chordal bipartite graphs , 1996, Discret. Math..

[41]  Pavel Pudlák,et al.  On the Computational Power of Depth-2 Circuits with Threshold and Modulo Gates , 1997, Theor. Comput. Sci..

[42]  Ileana Streinu,et al.  The vertex-edge visibility graph of a polygon , 1998, Comput. Geom..

[43]  Igor Walukiewicz,et al.  Relating Hierarchies of Word and Tree Automata , 1998, STACS.

[44]  Helmut Seidl,et al.  On distributive fixed-point expressions , 1999, RAIRO Theor. Informatics Appl..

[45]  HierarchyRan Raz,et al.  Separation of the Monotone NC , 1999 .

[46]  Klaus Jansen,et al.  On the Complexity of the Maximum Cut Problem , 1994, Nord. J. Comput..

[47]  David Manlove,et al.  The Hospitals/Residents Problem with Ties , 2000, SWAT.

[48]  Jeffrey Scott Vitter,et al.  External memory algorithms and data structures: dealing with massive data , 2001, CSUR.

[49]  Hartmut Klauck,et al.  Lower bounds for quantum communication complexity , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[50]  Matthias Krause.  On the Computational Power of Boolean Decision Lists , 2002, STACS.

[51]  A. Razborov.  Quantum communication complexity of symmetric predicates , 2002, quant-ph/0204025.

[52]  David Manlove,et al.  Strong Stability in the Hospitals/Residents Problem , 2003, STACS.

[53]  Dominique de Werra,et al.  On Split-Coloring Problems , 2005, J. Comb. Optim..

[54]  Lawrence H. Landweber,et al.  Decision problems forω-automata , 1969, Mathematical systems theory.

[55]  Orna Kupferman,et al.  Safraless decision procedures , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[56]  David R. Wood,et al.  On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane , 2005, Discret. Comput. Geom..

[57]  Florian Pfender.  Visibility graphs of point sets in the plane , 2006, Electron. Notes Discret. Math..

[58]  Mikkel Thorup,et al.  Time-space trade-offs for predecessor search , 2006, STOC '06.

[59]  Mikkel Thorup,et al.  Randomization does not help searching predecessors , 2007, SODA '07.

[60]  Subir Kumar Ghosh,et al.  Visibility Algorithms in the Plane , 2007 .

[61]  Zhiqiang Zhang,et al.  Communication complexities of symmetric XOR functions , 2009, Quantum Inf. Comput..

[62]  Troy Lee,et al.  Lower Bounds in Communication Complexity , 2009, Found. Trends Theor. Comput. Sci..

[63]  Christof Löding,et al.  Regular Cost Functions over Finite Trees , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[64]  Nancy A. Lynch,et al.  Distributed computation in dynamic networks , 2010, STOC '10.

[65]  Qin Zhang,et al.  The limits of buffering: a tight lower bound for dynamic membership in the external memory model , 2010, STOC '10.

[66]  Christian Komusiewicz,et al.  Measuring Indifference: Unit Interval Vertex Deletion , 2010, WG.

[67]  Mihai Patrascu,et al.  Towards polynomial lower bounds for dynamic problems , 2010, STOC '10.

[68]  Chien-Chung Huang,et al.  Classified stable matching , 2009, SODA '10.

[69]  Alexander A. Sherstov.  The unbounded-error communication complexity of symmetric functions , 2011, Comb..

[70]  Günter Rote,et al.  Convexifying Polygons Without Losing Visibilities , 2011, CCCG.

[71]  Fabian Kuhn,et al.  Dynamic networks: models and algorithms , 2011, SIGA.

[72]  Nir Shavit.  Data structures in the multicore age , 2011, CACM.

[73]  Kasper Green Larsen.  The cell probe complexity of dynamic range counting , 2011, STOC '12.

[74]  Fedor V. Fomin,et al.  Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[75]  Michal Pilipczuk,et al.  Largest Chordal and Interval Subgraphs Faster Than 2 n , 2013, ESA.

[76]  Telikepalli Kavitha,et al.  Popular matchings in the stable marriage problem , 2011, Inf. Comput..

[77]  L. S. Shapley,et al.  College Admissions and the Stability of Marriage , 2013, Am. Math. Mon..

[78]  Marcin Pilipczuk,et al.  Split Vertex Deletion meets Vertex Cover: New fixed-parameter and exact exponential-time algorithms , 2013, Inf. Process. Lett..

[79]  Fedor V. Fomin,et al.  Large induced subgraphs via triangulations and CMSO , 2014, SODA.

[80]  Shuichi Miyazaki,et al.  The Hospitals/Residents Problem with Lower Quotas , 2014, Algorithmica.

[81]  Matús Mihalák,et al.  Improved bounds for the conflict-free chromatic art gallery problem , 2014, SoCG.

[82]  Yixin Cao,et al.  Chordal Editing is Fixed-Parameter Tractable , 2014, STACS.

[83]  Shengyu Zhang.  Efficient quantum protocols for XOR functions , 2014, SODA.

[84]  TELIKEPALLI KAVITHA,et al.  A Size-Popularity Tradeoff in the Stable Marriage Problem , 2014, SIAM J. Comput..

[85]  Sándor P. Fekete,et al.  On the Chromatic Art Gallery Problem , 2014, CCCG.

[86]  Joshua Brody,et al.  Adapt or Die: Polynomial Lower Bounds for Non-Adaptive Dynamic Data Structures , 2015, Theory Comput..

[87]  M. Yamashita,et al.  DS-1-10 THE STRUCTURE OF POPULAR MATCHINGS IN A STABLE MARRIAGE PROBLEM , 2015 .

[88]  Telikepalli Kavitha,et al.  Popular Matchings with Two-Sided Preferences and One-Sided Ties , 2015, ICALP.

[89]  Yixin Cao,et al.  Interval Deletion Is Fixed-Parameter Tractable , 2012, SODA.

[90]  Allan Grønlund Jørgensen,et al.  New Unconditional Hardness Results for Dynamic and Online Problems , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[91]  Subhash Suri,et al.  Tight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Art Galleries , 2014, SoCG.

[92]  Matthew Babbitt,et al.  On k-visibility graphs , 2015, J. Graph Algorithms Appl..

[93]  Toniann Pitassi,et al.  Deterministic Communication vs. Partition Number , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[94]  Faith Ellen,et al.  Upper and Lower Bounds on the Power of Advice , 2016, SIAM J. Comput..

[95]  Omri Weinstein,et al.  Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[96]  Yixin Cao,et al.  Linear Recognition of Almost Interval Graphs , 2014, SODA.

[97]  Toniann Pitassi,et al.  Query-to-Communication Lifting for BPP , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[98]  Meghana Nasre,et al.  Popularity in the generalized Hospital Residents Setting , 2017, CSR.

[99]  Saket Saurabh,et al.  Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion , 2017, SODA.

[100]  Arkadev Chattopadhyay,et al.  Dual polynomials and communication complexity of XOR functions , 2017, Electron. Colloquium Comput. Complex..

[101]  Hamed Hatami,et al.  The Unbounded-Error Communication Complexity of symmetric XOR functions , 2017, ArXiv.

[102]  Anup Rao,et al.  Non-Adaptive Data Structure Lower Bounds for Median and Predecessor Search from Sunflowers , 2017, Electron. Colloquium Comput. Complex..

[103]  Telikepalli Kavitha,et al.  Popularity, Mixed Matchings, and Self-duality , 2017, SODA.

[104]  Yixin Cao,et al.  Unit interval editing is fixed-parameter tractable , 2015, Inf. Comput..

[105]  Yixin Cao,et al.  Approximate association via dissociation , 2017, Discret. Appl. Math..

[106]  Marcin Pilipczuk,et al.  Approximation and Kernelization for Chordal Vertex Deletion , 2016, SODA.

[107]  Michal Skrzypczak,et al.  How Deterministic are Good-For-Games Automata? , 2017, FSTTCS.

[108]  Fedor V. Fomin,et al.  Exact Algorithms via Monotone Local Search , 2019, J. ACM.

[109]  Ajit A. Diwan,et al.  On colouring point visibility graphs , 2020, Discret. Appl. Math..