arXiv subjects

Shayan Chashm Jahan

Publications and source records attributed to Shayan Chashm Jahan.

3 recordsLinked to original sources

arXiv2024

Bi-Criteria Metric Distortion

Selecting representatives based on voters' preferences is a fundamental problem in social choice theory. While cardinal utility functions offer a detailed representation of preferences, ordinal rankings are often the only available information due to their simplicity and practical constraints. The metric distortion framework addresses this issue by modeling voters and candidates as points in a metric space, with distortion quantifying the efficiency loss from relying solely on ordinal rankings. Existing works define the cost of a voter with respect to a candidate as their distance and set the overall cost as either the sum (utilitarian) or maximum (egalitarian) of these costs across all voters. They show that deterministic algorithms achieve a best-possible distortion of 3 for any metric when considering a single candidate. This paper explores whether one can obtain a better approximation compared to an optimal candidate by relying on a committee of $k$ candidates ($k \ge 1$), where the cost of a voter is defined as its distance to the closest candidate in the committee. We answer this affirmatively in the case of line metrics, demonstrating that with $O(1)$ candidates, it is possible to achieve optimal cost. Our results extend to both utilitarian and egalitarian objectives, providing new upper bounds for the problem. We complement our results with lower bounds for both the line and 2-D Euclidean metrics.

cs.GT↗

arXiv2024

Fairness and Efficiency in Online Class Matching

The online bipartite matching problem, extensively studied in the literature, deals with the allocation of online arriving vertices (items) to a predetermined set of offline vertices (agents). However, little attention has been given to the concept of class fairness, where agents are categorized into different classes, and the matching algorithm must ensure equitable distribution across these classes. We here focus on randomized algorithms for the fair matching of indivisible items, subject to various definitions of fairness. Our main contribution is the first (randomized) non-wasteful algorithm that simultaneously achieves a $1/2$ approximation to class envy-freeness (CEF) while simultaneously ensuring an equivalent approximation to the class proportionality (CPROP) and utilitarian social welfare (USW) objectives. We supplement this result by demonstrating that no non-wasteful algorithm can achieve an $α$-CEF guarantee for $α> 0.761$. In a similar vein, we provide a novel input instance for deterministic divisible matching that demonstrates a nearly tight CEF approximation. Lastly, we define the ``price of fairness,'' which represents the trade-off between optimal and fair matching. We demonstrate that increasing the level of fairness in the approximation of the solution leads to a decrease in the objective of maximizing USW, following an inverse proportionality relationship.

cs.GT↗

arXiv2023

Rainbow Cycle Number and EFX Allocations: (Almost) Closing the Gap

Recently, some studies on the fair allocation of indivisible goods notice a connection between a purely combinatorial problem called the Rainbow Cycle problem and a fairness notion known as $\efx$: assuming that the rainbow cycle number for parameter $d$ (i.e. $\rainbow(d)$) is $O(d^β\log^γd)$, we can find a $(1-ε)$-$\efx$ allocation with $O_ε(n^{\fracβ{β+1}}\log^{\fracγ{β+1}} n)$ number of discarded goods \cite{chaudhury2021improving}. The best upper bound on $\rainbow(d)$ is improved in a series of works to $O(d^4)$ \cite{chaudhury2021improving}, $O(d^{2+o(1)})$ \cite{berendsohn2022fixed}, and finally to $O(d^2)$ \cite{Akrami2022}.\footnote{We refer to the note at the end of the introduction for a short discussion on the result of \cite{Akrami2022}.} Also, via a simple observation, we have $\rainbow(d) \in Ω(d)$ \cite{chaudhury2021improving}. In this paper, we introduce another problem in extremal combinatorics. For a parameter $\ell$, we define the rainbow path degree and denote it by $\ech(\ell)$. We show that any lower bound on $\ech(\ell)$ yields an upper bound on $\rainbow(d)$. Next, we prove that $\ech(\ell) \in Ω(\ell^2/\log n)$ which yields an almost tight upper bound of $\rainbow(d) \in Ω(d \log d)$. This in turn proves the existence of $(1-ε)$-$\efx$ allocation with $O_ε(\sqrt{n \log n})$ number of discarded goods. In addition, for the special case of the Rainbow Cycle problem that the edges in each part form a permutation, we improve the upper bound to $\rainbow(d) \leq 2d-4$. We leverage $\ech(\ell)$ to achieve this bound. Our conjecture is that the exact value of $\ech(\ell) $ is $ \lfloor \frac{\ell^2}{2} \rfloor -1$. We provide some experiments that support this conjecture. Assuming this conjecture is correct, we have $\rainbow(d) \in Θ(d)$.

cs.GT↗