A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

• Published in: The Journal of artificial intelligence research Vol. 50; pp. 409 - 446
• Main Authors: Bredereck, R., Chen, J., Hartung, S., Kratsch, S., Niedermeier, R., Suchy, O., Woeginger, G. J.
• Format: Journal Article
• Language: English
• Published: San Francisco AI Access Foundation 01.01.2014
• Subjects: Artificial intelligence; Complexity; Greedy algorithms; Lobbying; Parameter modification; Voters
• ISSN: 1076-9757; 1076-9757; 1943-5037
• DOI: 10.1613/jair.4285

Assume that each of n voters may or may not approve each of m issues. If an agent (the lobby) may influence up to k voters, then the central question of the NP-hard Lobbying problem is whether the lobby can choose the voters to be influenced so that as a result each issue gets a majority of approvals. This problem can be modeled as a simple matrix modification problem: Can one replace k rows of a binary n x m-matrix by k all-1 rows such that each column in the resulting matrix has a majority of 1s? Significantly extending on previous work that showed parameterized intractability (W[2]-completeness) with respect to the number k of modified rows, we study how natural parameters such as n, m, k, or the "maximum number of 1s missing for any column to have a majority of 1s" (referred to as "gap value g") govern the computational complexity of Lobbying. Among other results, we prove that Lobbying is fixed-parameter tractable for parameter m and provide a greedy logarithmic-factor approximation algorithm which solves Lobbying even optimally if m < 5. We also show empirically that this greedy algorithm performs well on general instances. As a further key result, we prove that Lobbying is LOGSNP-complete for constant values g>0, thus providing a first natural complete problem from voting for this complexity class of limited nondeterminism.