A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

Abstract

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×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 ≤ 4. 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 ≥ 1, thus providing a first natural complete problem from voting for this complexity class of limited nondeterminism.

Lossy kernelization

“…We extend the machinery for showing kernelization lower bounds to the setting of approximate kernels, and use these new methods to prove lower bounds on the size of approximate kernels for Longest Path parameterized by the objective function value, and Set Cover and Hitting Set parameterized by universe size. Especially Set Cover parameterized by universe size has been a useful starting point for reductions showing lower bounds for traditional kernelization [10,18,22,31,34,46,56]. We are therefore confident that our work lays a solid foundation for future work on approximate kernelization.…”

Stable Marriage with Multi-Modal Preferences

“…LOGSNP-hardness has been encountered and discussed for natural problems in Computational Social Choice[10,11].…”

Complexity of manipulation and bribery in judgment aggregation for uniform premise-based quota rules

“…In particular, Christian et al (2007) (see also, e.g., Bredereck et al, 2012;Binkele-Raible et al, 2014) studied the related problem of optimal lobbying, which may be seen as a simplified variant of judgment aggregation. We will apply their hardness result for optimal lobbying in the proof of Theorem 17 in Section 4 when we will be concerned with bribery in judgment aggregation.…”