Abstract
We investigate issues regarding two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [2] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant—optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm.
The problem of determining Dodgson winners is known to be complete for parallel access to NP [11]. Homan and Hemaspaandra [10] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is a “frequently self-knowingly correct algorithm.” We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein’s junta distributions [15].
Supported in part by DFG grants RO 1202/9-1 and RO 1202/9-3, NSF grant CCF-0426761, the Alexander von Humboldt Foundation’s TransCoop program, and a Friedrich Wilhelm Bessel Research Award. Work done in part while the second author was visiting Heinrich-Heine-Universität Düsseldorf. Some of the results of Section [3] of this paper were presented at the First International Workshop on Computational Social Choice, December 2006.
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Erdélyi, G., Hemaspaandra, L.A., Rothe, J., Spakowski, H. (2007). On Approximating Optimal Weighted Lobbying, and Frequency of Correctness Versus Average-Case Polynomial Time. In: Csuhaj-Varjú, E., Ésik, Z. (eds) Fundamentals of Computation Theory. FCT 2007. Lecture Notes in Computer Science, vol 4639. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74240-1_26
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