A Lower Bound for Local Search Proportional Approval Voting

TL;DR

This work resolves an open question about the polynomial-time convergence of bounded local search for Proportional Approval Voting when the improvement threshold $\varepsilon$ is arbitrarily small. It proves a super-polynomial lower bound, $\Omega(k^{\log k})$, on the length of improving sequences for $0^+$-ls-PAV under adversarial better-response, by constructing a polynomial-size election built from hierarchical building blocks $E(j,k)$ and their aggregations $E^t(j,k)$. The result extends to a fixed lexicographic pivoting rule, showing that even with a deterministic swap order, the process may take super-polynomial time, while leaving open the best-response variant. The paper also provides empirical comparisons (in the extended version) suggesting that better-response can be faster in practice, and argues that these findings justify the previously used threshold $\varepsilon=\frac{n}{k^2}$ to balance fairness and tractability.

Abstract

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 $\sum_{j=1}^t\frac{1}{j}$ 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 by Aziz et al. It proceeds by starting with an arbitrary size-$k$ set $W$ and, at each step, checking if there is a pair of candidates $a\in W$, $b\not\in W$ such that swapping $a$ and $b$ increases the total welfare by at least $\varepsilon$; if yes, it performs the swap. Aziz et al.~show that setting $\varepsilon=\frac{n}{k^2}$ ensures both the desired fairness guarantees and polynomial running time. However, they leave it open whether the algorithm converges in polynomial time if $\varepsilon$ 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 $\varepsilon$ 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.

Figures (3)

• Figure 1: For $k=3$ and initial committee $W=\{c_1,c_2,c_3\}$ , $\frac{n}{k^2}$-ls-PAV would replace a member of $W$ with one of $c_4,c_5$ in the instance in Figure \ref{['sixtythirty']}, but not in the instance in Figure \ref{['noise']}.
• Figure 2: Highlighted candidates are in the committee; arrows from $a$ to $b$ labelled with $t$ indicate that $a$ is replaced by $b$ in iteration $t$. Observe that pink candidates are only replaced by pink candidates; similarly, blue candidates are only replaced by blue candidates. This illustrates a swap sequence similar to that in the proof of Theorem \ref{['thm:threshold']}, with the exception that, to make the figure visually appealing, we display an equal number of blue and pink candidates.
• Figure 3: We illustrate the sequence of swaps in Theorem \ref{['thm:zerolow']} by a small example. Each one of the $23$$3\times 4$ grids shows 12 candidates, one for each empty square, the 4 coloured ones indicating candidates currently in the committee. The empty squares in column $i$ are the candidates in $C_i$; they are ordered as $c_{i,1},c_{i,2},c_{i,3}$ from top to bottom. We omit the dummy candidates from this picture, and let $k_1=4$ and $t+1=3$, as larger $t$ is only necessary for the sequence length in the proof. An arrow indicates the swap that will transform the current committee into the next committee. The top left initial committee is $\{c_{1,3},c_{2,3},c_{3,3},c_{4,1}\}$ and the bottom right final committee is $\{c_{1,3},c_{2,3},c_{3,3},c_{4,3}\}$.

Theorems & Definitions (9)

• Conjecture 1: left open by AE+18
• Theorem 2
• Theorem 3
• Proposition 4
• Lemma 4
• Corollary 5
• Proposition 6
• Proposition 6
• Theorem 7