arXiv subjects

Ulysse Hennebelle

Publications and source records attributed to Ulysse Hennebelle.

3 recordsLinked to original sources

arXiv2026

Condorcet-Winning Sets and Peer Selection in Planar Metric Elections

In ranked-choice voting, a Condorcet-winning set is a group of candidates for which no outside candidate is preferred to every member of the group by a majority of voters. We study Condorcet-winning sets in planar metric elections, where voters rank candidates according to their distance under a given norm. We formulate general metric elections, peer selection, and the one-round Voronoi game as instances of a two-player Stackelberg game and place these problems in a common hierarchy. We also introduce a new variant, which we call strong peer selection. Our main result concerns peer selection under the $\ell_1$ and $\ell_\infty$ norms. We prove that every planar instance admits a Condorcet-winning set of size at most three, even under strong peer selection. This follows from a new result for strong rectangular $\varepsilon$-nets. We show that, for every set of points in the plane, one can choose at most three input points that intersect every axis-parallel rectangle containing more than half of the points, improving the previous threshold of $9 / 16$ due to Ashok et al. Under the $\ell_2$ norm, we prove that every planar metric election admits a Condorcet-winning set of size at most four, improving on the general bound of five due to Song et al. Finally, we give new norm-independent bounds for the one-round Voronoi game.

cs.GT↗

arXiv2026

Two-sided Assortment Optimization: Adaptivity Gaps and Approximation Algorithms

To address efficiency and design challenges in choice-based matching platforms, we introduce a two-sided assortment optimization framework under general choice preferences. The goal in this problem is to maximize the expected number of matches by deciding which assortments are displayed to the agents and the order in which they are shown. In this context, we identify several classes of policies that platforms can use in their design. Our goals are: (1) to measure the value that one class of policies has over another one, and (2) to approximately solve the optimization problem itself for a given class. For (1), we define the adaptivity gap as the worst-case ratio between the optimal values of two different policy classes. First, we show that the gap between the class of policies that statically show assortments to one-side first and the class of policies that adaptively show assortments to one-side first is exactly $e/(e-1)$. Second, we show that the gap between the latter class of policies and the fully adaptive class of policies that show assortments to agents one by one is exactly $2$. We also note that the worst policies are those who simultaneously show assortments to all the agents. For (2), we first design a polynomial time algorithm that achieves a $1/4$ approximation factor within the class of policies that adaptively show assortments to agents one by one. Furthermore, when agents' preferences are governed by multinomial-logit models, we show that a 0.067 approximation factor can be obtained within the class of policies that show assortments to all agents at once. We further generalize our results to constrained assortment settings, where we impose an upper bound on the size of the displayed assortments. Finally, we present a computational study to evaluate the empirical performance of our theoretical guarantees.

math.OC↗

arXiv2023

Runtime Analyses of Multi-Objective Evolutionary Algorithms in the Presence of Noise

In single-objective optimization, it is well known that evolutionary algorithms also without further adjustments can tolerate a certain amount of noise in the evaluation of the objective function. In contrast, this question is not at all understood for multi-objective optimization. In this work, we conduct the first mathematical runtime analysis of a simple multi-objective evolutionary algorithm (MOEA) on a classic benchmark in the presence of noise in the objective functions. We prove that when bit-wise prior noise with rate $p \le α/n$, $α$ a suitable constant, is present, the \emph{simple evolutionary multi-objective optimizer} (SEMO) without any adjustments to cope with noise finds the Pareto front of the OneMinMax benchmark in time $O(n^2\log n)$, just as in the case without noise. Given that the problem here is to arrive at a population consisting of $n+1$ individuals witnessing the Pareto front, this is a surprisingly strong robustness to noise (comparably simple evolutionary algorithms cannot optimize the single-objective OneMax problem in polynomial time when $p = ω(\log(n)/n)$). Our proofs suggest that the strong robustness of the MOEA stems from its implicit diversity mechanism designed to enable it to compute a population covering the whole Pareto front. Interestingly this result only holds when the objective value of a solution is determined only once and the algorithm from that point on works with this, possibly noisy, objective value. We prove that when all solutions are reevaluated in each iteration, then any noise rate $p = ω(\log(n)/n^2)$ leads to a super-polynomial runtime. This is very different from single-objective optimization, where it is generally preferred to reevaluate solutions whenever their fitness is important and where examples are known such that not reevaluating solutions can lead to catastrophic performance losses.

cs.NE↗