arXiv subjects

Yeeseok Oh

Publications and source records attributed to Yeeseok Oh.

4 recordsLinked to original sources

arXiv2026

Graph Coloring with Color Preferences

We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its original one. We define the stable chromatic number $\chi_\mathrm{stable}(G)$ of a graph $G$ as the minimum integer $k$ such that every preference profile admits a stable $k$-coloring of $G$. We establish several upper and lower bounds. In particular, for any acyclic orientation of the edges of $G$, the largest number of vertices reachable from a vertex by directed paths, including the vertex itself, is an upper bound on $\chi_\mathrm{stable}(G)$. This shows that $\chi_\mathrm{stable}(G)$ is well-defined. We also show that $O(t \log (1+n/t))$ colors suffice for an $n$-vertex graph $G$ of treewidth $t$, and complement this with a lower bound in terms of the Grundy number. Turning to the problem of finding a minimum stable coloring for a given profile, we show that stable $2$-colorability is polynomial-time solvable, whereas stable $k$-colorability is NP-complete for every fixed $k\ge 3$. Using the treewidth bound, we give a fixed-parameter tractable algorithm parameterized by treewidth.

cs.GT↗

arXiv2026

Optimally Selecting Representative Agents from a Metric Space

This paper studies the problem of proportionally fair clustering, where the goal is to select $k$ ``centers'' from a metric space that fairly represent a set of agents who also lie in the metric space. Specifically, we focus on finding a clustering satisfying a fairness property known as the Droop core. In the practical special case in which the set of feasible center locations contains every agent location, the previous best-known result guaranteed a $(1 + \sqrt{2})$-approximation of the Droop core, while the best-known lower bound was $2$. In this paper, we show that this lower bound is tight and that a clustering in the $2$-Droop core always exists. Further, we show that such a clustering can be achieved by only selecting centers from locations in the metric space where an agent resides. We establish this using Scarf's theorem guaranteeing a nonempty core for balanced non-transferable utility games. This result has several interesting corollaries. Most notably, it resolves the $β$-plurality problem of Aronov et al. [2021] for general metric spaces. The main result of this paper was generated by $\mathtt{ChatGPT}$-$\mathtt{5.6}$-$\mathtt{Sol}$ through a series of interactions with the authors. The authors of this paper verified the generated proof and rewrote it for clarity.

cs.GT↗

arXiv2026

Candidate Resignation Monotonicity in Approval-Based Committee Elections

Approval-based committee (ABC) elections select a fixed-size committee of candidates based on voters' approval preferences. We study a setting where elected members may resign and formalize resignation monotonicity: when we re-run the voting rule after resignations, a resignation monotone rule must still elect all remaining previously winning candidates. We show that many prominent ABC rules fail this property, and by connecting our notion to prior work on ABC elections with dynamic candidate sets, we show that the justified representation (JR) axiom is incompatible with resignation monotonicity. It turns out that fractional committee voting does not suffer from this issue: rules based on using maximum flows in the network representation of an election instance can satisfy both representation axioms and resignation monotonicity. We design an integral version of maximum flow, the Maximum Payment Rule (MPR), and show that it satisfies a relaxed form of resignation monotonicity that only requires there to be some way of replacing the resigning candidates in a way that guarantees the PJR+ axiom. MPR is NP-hard to compute in general, but becomes tractable in structured domains. Finally, we study a strategic setting where a losing candidate may introduce new weak candidates to the election to try to become a winner. We show that all resignation monotone rules as well as many sequential rules are immune, but that the PAV rule can be manipulated in this way.

cs.GT↗

arXiv2026

Improved Lower Bounds for Proportionally Fair Clustering

We study proportionally fair clustering, where a set of $k$ centers must be chosen from a metric space to represent $n$ agents, and no sufficiently large group of agents should be collectively underrepresented. One of the central notions of fairness in this setting is the $α$-core. The existence of clusterings in the $(1+\sqrt{2})$-core was established by Chen et al. [2019], who also showed instances where the $α$-core is empty for every $α< 2$. Closing this gap has remained an open problem for seven years. We make progress from the lower-bound side by providing an instance whose $α$-core is empty for every $α< 2.1508$. Our techniques rely on establishing connections between variants of the core, namely the Hare core and the Droop core; reducing the search for optimal empty-core instances to a highly structured family of clustering instances; and using a Mixed Integer Linear Program (MILP) to search for optimal lower-bound instances within this reduced space. Using this framework, we also determine tight bounds for Droop quota clustering instances with a small number of possible candidate centers and a single center to be selected. For each number of centers $m \in \{3,4,5,6\}$, we give the exact threshold $α_m^*$ such that an $α_m^*$-core clustering always exists, while for every $α< α_m^*$ there is an instance with $m$ centers whose $α$-core is empty. Although these values were originally found through computer-aided search, we also provide direct proofs that do not rely on MILP certificates.

cs.GT↗