arXiv subjects

Agnès Totschnig

Publications and source records attributed to Agnès Totschnig.

5 recordsLinked to original sources

arXiv2026

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↗

arXiv2026

The Complexity of Justified Representation with Additive Utilities

We study the computational complexity of satisfying proportional representation -- in particular proportional, extended, and fully justified representation (PJR, EJR, and FJR) -- in participatory budgeting and committee elections with additive utilities. First, we give a complete picture of the complexity of the axioms for a constant number of voters or voter types. Second, we show that even for committee elections with integer utilities bounded above by a small constant, satisfying FJR is intractable, giving the first strong NP-hardness result for a justified representation axiom. Third, we extend the Expanding Approvals Rule to committee elections with additive utilities and show that it satisfies PJR. Lastly, we show that no sequential voting rule can improve on the known positive result, thus proving that novel, substantially different voting rules are needed to surpass these boundaries. Beyond their theoretical merit, our results carry practical importance, as multi-winner voting with additive utilities has recently been gaining prominence in online deliberation and real-world participatory budgeting.

cs.GT↗

arXiv2026

Almost perfect graph classes

A graph $G$ is perfect if $ω(H) = χ(H)$ for each induced subgraph $H$ of $G$. In 2002, Chudnovsky, Robertson, Seymour, and Thomas famously proved the Strong Perfect Graph Theorem. Motivated by this forbidden induced subgraph characterization of the class of perfect graphs as well as the possible extension of efficient algorithms on perfect graphs, we consider the structure of graphs that are almost perfect. We say a graph is $c$-apex perfect if there is a constant $c$ number of vertices such that, upon the deletion of these vertices, what remains is a perfect graph. In this paper, we characterize the class of the sets of graphs $\mathcal{H}$ with $|\mathcal{H}|\leq 2$ for which there exists $c \in \mathbb{N}$ with the property that each $\mathcal{H}$-free graph is $c$-apex perfect. We also extend these results to several notable subclasses of perfect graphs, including chordal, interval, split, bipartite, and complete multipartite graphs.

math.CO↗

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 Bilò 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

The Tiered Clinching Auction with Applications to Carbon Offset Markets

Voluntary carbon offsetting is a strategy which has been pursued globally by corporations to reduce their effective carbon emissions. Carbon offset markets currently suffer from low liquidity: offsets are highly heterogeneous, the landscape of accreditation is fragmented, and the market lacks a centralized trading infrastructure which would provide clear demand and price signaling to producers and buyers of offsets. In this paper, we propose an ascending auction mechanism which can be applied to the sale of carbon offsets. It generalizes Ausubel's clinching auction to a setting with items of tiered quality. The central idea behind the clinching mechanism is that bidders are allocated items when their opponents' demand drops below the supply. This is generalized to multiple nested submarkets where clinching can occur in the tiered case. Assuming that bidders have a minimum quality level that they will accept but are indifferent to quality beyond that, along with having decreasing marginal utility, we obtain that the auction generates the efficient outcome and charges VCG prices. As a result, sincere bidding is a weakly dominant strategy given private values. Moreover, the auction can be implemented in polynomial time. Beyond offset markets, this auction can be applied to any market where items can be ordered on a scale and bidders have a cutoff point for desiring items on the scale, such as hotel room bookings, concert ticketing and sponsored search auctions.

cs.GT↗