arXiv subjects

Frank Connor

Publications and source records attributed to Frank Connor.

5 recordsLinked to original sources

arXiv2026

Learning Proportional Committees from Violation Feedback

We study violation-feedback learning of proportionally representative approval-based committees. In each round, a learner proposes a committee of size $k$. An oracle either accepts the proposal or adversarially selects a representation violation with respect to a single fixed hidden approval profile. We compare \emph{full-witness feedback}, which reveals the violation level, an omitted candidate, and the affected voter group, with \emph{candidate-only feedback}, which reveals only that candidate. The target notions are proportional justified representation plus (PJR+) and extended justified representation plus (EJR+). In every setting we study, the number of rejected proposals can be bounded solely in terms of $k$, with no dependence on the numbers of voters and candidates. For PJR+, the optimal deterministic and randomized rejection complexities equal $k$ under both feedback models. For EJR+, the picture is more nuanced. Under full-witness feedback, we prove an $\Omega(k^{3/2})$ deterministic lower bound and give a deterministic polynomial-time algorithm using $O(k^2\log k)$ rejections. Under candidate-only feedback, randomization achieves $O(k^2\log k)$ expected rejections via uniform random deletion, while deterministic exhaustive branching gives a $2^{O(k^2(\log k)^2)}$ rejection bound. Even with full-witness feedback, randomized learners may require $k$ rejections.

cs.GT↗

arXiv2026

Fair Division of Graphs: Beyond Traceability

In this paper, we study fair division problems in which resources are structured as graphs and agents must receive connected bundles. This connectivity requirement fundamentally alters the problem, making it significantly more challenging than its classical counterpart. We focus on the fairness notion of $\mathrm{EF1}_{\mathrm{outer}}$, where envy can be eliminated by removing at most one vertex whose deletion does not disconnect the bundle -- a critical constraint for applications such as land division and network allocation. Our first result extends prior work by establishing the existence of $\mathrm{EF1}_{\mathrm{outer}}$ allocations for an infinite family of non-traceable graphs (that is, graphs that do not admit a Hamiltonian path), answering a central open question and generalizing Bil\`o et al.'s result for traceable graphs. We then make progress on a conjecture concerning the $\mathrm{EF1}_{\mathrm{outer}}$ spectrum of trees due to Chen and Zwicker. Finally, we complement our structural results with algorithmic insights, showing that deciding the existence of an $\mathrm{EF1}_{\mathrm{outer}}$ allocation is NP-complete even for binary additive valuations, thereby resolving an open complexity question. Taken together, our results deepen the connection between graph theory and fair division, and offer new tools for studying fairness in structured resource environments.

cs.GT↗

arXiv2026

Random Serial Dictatorship is $\sqrt{2}$-Envy-Free

We analyze the house allocation problem, in which a set of agents must be matched to a set of objects for which they have cardinal utilities. A central mechanism for this problem is random serial dictatorship (RSD), which has long served as a canonical subject of study due to its simplicity and the existence of exact characterizations by its properties. Despite this extensive understanding, a basic quantitative question about the fairness of this mechanism remains unresolved. Although RSD is often viewed as fair ex ante, surprisingly, it is not envy-free in expectation. We quantify its deviation from envy-freeness via the envy-ratio, which is the maximum over all instances and pairs of agents of the ratio between an agent's expected utility for another agent's random object and for its own random object. Prior work shows a factor-$\sqrt{2}\approx 1.414$ lower bound on the envy-ratio of RSD. Our headline result is a matching upper bound, showing that RSD is $\sqrt{2}$-envy-free in the house allocation problem. We further analyze the two natural extensions of RSD (the randomized round-robin mechanism and the iterated-RSD mechanism) to settings with unequal numbers of agents and objects and more general valuation classes. For additive valuations, this ratio increases to at least $1.5$ and at most $1.707$ for randomized round-robin, but remains exactly $\sqrt{2}$ for iterated-RSD. For submodular valuations, we prove constant-factor upper and lower bounds for both mechanisms, leaving only a small constant gap in both cases. For the more general classes of XOS and subadditive valuations, we present a tight analysis for both mechanisms, showing that the envy-ratio is unbounded in the number of agents. These results provide the first tight or nearly tight quantitative guarantees on the extent to which random serial dictatorship and its natural generalizations approximate envy-freeness.

cs.GT↗

arXiv2026

Tight Asymptotic Bounds for Fair Division With Externalities

We study the problem of allocating a set of indivisible items among agents whose preferences include externalities. Unlike the standard fair division model, agents may derive positive or negative utility not only from items allocated directly to them, but also from items allocated to other agents. Since exact envy-freeness cannot be guaranteed, prior work has focused on its relaxations. However, two central questions remained open: does there always exist an allocation that is envy-free up to one item (EF1), and if not, what is the optimal relaxation EF-$k$ that can always be attained? We settle both questions by deriving tight asymptotic bounds on the number of items sufficient to eliminate envy. We show that for any instance with $n$ agents, an allocation that is envy-free up to $O(\sqrt{n})$ items always exists and can be found in polynomial time, and we prove a matching $\Omega(\sqrt{n})$ lower bound showing that this result is tight even for binary valuations, which rules out the existence of EF1 allocations when agents have externalities.

cs.GT↗

arXiv2025

The Popular Dimension of Matchings

We study popular matchings in three classical settings: the house allocation problem, the marriage problem, and the roommates problem. In the popular matching problem, (a subset of) the vertices in a graph have preference orderings over their potential matches. A matching is popular if it gets a plurality of votes in a pairwise election against any other matching. Unfortunately, popular matchings typically do not exist. So we study a natural relaxation, namely popular winning sets which are a set of matchings that collectively get a plurality of votes in a pairwise election against any other matching. The $\textit{popular dimension}$ is the minimum cardinality of a popular winning set, in the worst case over the problem class. We prove that the popular dimension is exactly $2$ in the house allocation problem, even if the voters are weighted and ties are allowed in their preference lists. For the marriage problem and the roommates problem, we prove that the popular dimension is between $2$ and $3$, when the agents are weighted and/or their preferences orderings allow ties. In the special case where the agents are unweighted and have strict preference orderings, the popular dimension of the marriage problem is known to be exactly $1$ and we prove the popular dimension of the roommates problem is exactly $2$.

cs.GT↗