Large-Scale Election Campaigns: Combinatorial Shift Bribery

We study the complexity of a combinatorial variant of the Shift Bribery problem in elections. In the standard Shift Bribery problem, we are given an election where each voter has a preference order over the candidate set and where an outside agent, the briber, can pay each voter to rank the briber's favorite candidate a given number of positions higher. The goal is to ensure the victory of the briber's preferred candidate. The combinatorial variant of the problem, introduced in this paper, models settings where it is possible to affect the position of the preferred candidate in multiple votes, either positively or negatively, with a single bribery action. This variant of the problem is particularly interesting in the context of large-scale campaign management problems (which, from the technical side, are modeled as bribery problems). We show that, in general, the combinatorial variant of the problem is highly intractable (NP-hard, hard in the parameterized sense, and hard to approximate), but we provide some (approximation) algorithms for natural restricted cases.

[1]  P. Faliszewski,et al.  Control and Bribery in Voting , 2016, Handbook of Computational Social Choice.

[2]  David Cary,et al.  Estimating the Margin of Victory for Instant-Runoff Voting , 2011, EVT/WOTE.

[3]  Michael R. Fellows,et al.  Fundamentals of Parameterized Complexity , 2013 .

[4]  Vincent Conitzer,et al.  How hard is it to control sequential elections via the agenda , 2009, IJCAI 2009.

[5]  Jörg Rothe,et al.  The complexity of probabilistic lobbying , 2014, Discret. Optim..

[6]  Piotr Faliszewski,et al.  Campaign Management Under Approval-Driven Voting Rules , 2011, Algorithmica.

[7]  Piotr Faliszewski,et al.  Prices matter for the parameterized complexity of shift bribery , 2014, Inf. Comput..

[8]  Lirong Xia,et al.  Computing the margin of victory for various voting rules , 2012, EC '12.

[9]  Piotr Faliszewski,et al.  Elections with Few Candidates: Prices, Weights, and Covering Problems , 2015, ADT.

[10]  Toby Walsh,et al.  How Hard Is It to Control an Election by Breaking Ties? , 2013, ECAI.

[11]  Felix Brandt,et al.  It only takes a few: on the hardness of voting with a constant number of agents , 2013, AAMAS.

[12]  Vijay V. Vazirani,et al.  Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs , 1999, SIAM J. Comput..

[13]  Piotr Faliszewski,et al.  Complexity of Shift Bribery in Committee Elections , 2016, AAAI.

[14]  Piotr Faliszewski,et al.  Campaigns for lazy voters: truncated ballots , 2012, AAMAS.

[15]  Sarit Kraus,et al.  How to Change a Group's Collective Decision? , 2013, IJCAI.

[16]  Piotr Faliszewski,et al.  Swap Bribery , 2009, SAGT.

[17]  Piotr Faliszewski,et al.  How Hard Is Bribery in Elections? , 2006, J. Artif. Intell. Res..

[18]  Ildikó Schlotter,et al.  Multivariate Complexity Analysis of Swap Bribery , 2010, Algorithmica.

[19]  Rolf Niedermeier,et al.  A Multivariate Complexity Analysis of Lobbying in Multiple Referenda , 2012, AAAI.

[20]  Piotr Faliszewski,et al.  Complexity of manipulation, bribery, and campaign management in Bucklin and fallback voting , 2013, Autonomous Agents and Multi-Agent Systems.

[21]  Rolf Niedermeier,et al.  Studies in Computational Aspects of Voting - A Parameterized Complexity Perspective , 2012, The Multivariate Algorithmic Revolution and Beyond.

[22]  Lirong Xia,et al.  Voting in Combinatorial Domains , 2016, Handbook of Computational Social Choice.

[23]  Stefan Szeider,et al.  Editing graphs to satisfy degree constraints: A parameterized approach , 2012, J. Comput. Syst. Sci..

[24]  Gerhard J. Woeginger,et al.  Multigraph realizations of degree sequences: Maximization is easy, minimization is hard , 2008, Oper. Res. Lett..

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

[26]  Andrew Klapper,et al.  On the complexity of bribery and manipulation in tournaments with uncertain information , 2012, J. Appl. Log..

[27]  Harold N. Gabow,et al.  An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems , 1983, STOC.

[28]  Piotr Faliszewski,et al.  Elections with Few Voters: Candidate Control Can Be Easy , 2014, AAAI.

[29]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[30]  Svetlana Obraztsova,et al.  On the Complexity of Voting Manipulation under Randomized Tie-Breaking , 2011, IJCAI.

[31]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[32]  Piotr Faliszewski,et al.  Combinatorial voter control in elections , 2014, Theor. Comput. Sci..

[33]  Ronen I. Brafman,et al.  CP-nets: A Tool for Representing and Reasoning withConditional Ceteris Paribus Preference Statements , 2011, J. Artif. Intell. Res..

[34]  Jörg Rothe,et al.  The Margin of Victory in Schulze, Cup, and Copeland Elections: Complexity of the Regular and Exact Variants , 2014, STAIRS.

[35]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[36]  Michael R. Fellows,et al.  On complexity of lobbying in multiple referenda , 2006 .

[37]  Ronald L. Rivest,et al.  Computing the Margin of Victory in IRV Elections , 2011, EVT/WOTE.

[38]  Svetlana Obraztsova,et al.  Ties Matter: Complexity of Voting Manipulation Revisited , 2011, IJCAI.

[39]  Piotr Faliszewski,et al.  Approximation Algorithms for Campaign Management , 2010, WINE.

[40]  Dimitrios M. Thilikos,et al.  Invitation to fixed-parameter algorithms , 2007, Comput. Sci. Rev..

[41]  Piotr Faliszewski,et al.  Using complexity to protect elections , 2010, Commun. ACM.

[42]  Maria Silvia Pini,et al.  Bribery in Voting Over Combinatorial Domains Is Easy , 2012, ISAIM.

[43]  Piotr Faliszewski,et al.  Llull and Copeland Voting Computationally Resist Bribery and Constructive Control , 2009, J. Artif. Intell. Res..