Selecting k out of m items based on the preferences of n heterogeneous agents is a widely studied problem in algorithmic game theory. If agents have approval preferences over individual items and harmonic utility functions over bundles - an agent receives ∑_{j = 1}^t1/j utility if t of her approved items are selected - then welfare optimisation is captured by a voting rule known as Proportional Approval Voting (PAV). PAV also satisfies demanding fairness axioms. However, finding a winning set of items under PAV is NP-hard. In search of a tractable method with strong fairness guarantees, a bounded local search version of PAV was proposed [Aziz et al., 2018]. It proceeds by starting with an arbitrary size-k set W and, at each step, checking if there is a pair of candidates a ∈ W, b ̸ ∈ W such that swapping a and b increases the total welfare by at least ε; if yes, it performs the swap. Aziz et al. show that setting ε = n/(k²) ensures both the desired fairness guarantees and polynomial running time. However, they leave it open whether the algorithm converges in polynomial time if ε is very small (in particular, if we do not stop until there are no welfare-improving swaps). We resolve this open question, by showing that if ε can be arbitrarily small, the running time of this algorithm may be super-polynomial. Specifically, we prove a lower bound of Ω(k^{log k}) if improvements are chosen lexicographically. To complement our lower bound, we provide an empirical comparison of two variants of local search - better-response and best-response - on several real-life data sets and a variety of synthetic data sets. Our experiments indicate that, empirically, better response exhibits faster running time than best response.
@InProceedings{kraiczy_et_al:LIPIcs.ESA.2024.82, author = {Kraiczy, Sonja and Elkind, Edith}, title = {{A Lower Bound for Local Search Proportional Approval Voting}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {82:1--82:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.82}, URN = {urn:nbn:de:0030-drops-211538}, doi = {10.4230/LIPIcs.ESA.2024.82}, annote = {Keywords: Computational Social Choice, Committee Elections, Local Search, Fairness} }
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