Andrei Constantinescu

In stable matching, one must find a matching between two sets of agents, commonly men and women, or job applicants and job positions. Each agent has a preference ordering over who they want to be matched with. Moreover a matching is said to be stable if no pair of agents prefer each other over their current matching. We consider solving stable matching in a distributed synchronous setting, where each agent is its own process. Moreover, we assume up to \(t_L\) agents on one side and \(t_R\) on the other side can be byzantine. After properly defining the stable matching problem in this setting, we study its solvability. When there are as many agents on each side with fully-ordered preference lists, we give necessary and sufficient conditions for stable matching to be solvable in the synchronous setting. These conditions depend on the communication model used, i.e., if parties on the same side are allowed to communicate directly, and on the presence of a cryptographic setup, i.e., digital signatures.

We introduce the Byzantine Selection Problem, living at the intersection of game theory and fault-tolerant distributed computing. Here, an event organizer is presented with a group of \(n\) agents, and wants to select \(\ell < n\) of them to form a team. For these purposes, each agent \(i\) self-reports a positive skill value \(v_i\), and a team's value is the sum of its members' skill values. Ideally, the value of the team should be as large as possible, which can be easily achieved by selecting agents with the highest \(\ell\) skill values. However, an unknown subset of at most \(t < n\) agents are byzantine and hence not to be trusted, rendering their true skill values as \(0\). In the spirit of the distributed computing literature, the identity of the byzantine agents is not random but instead chosen by an adversary aiming to minimize the value of the chosen team. Can we still select a team with good guarantees in this adversarial setting? As it turns out, deterministically, it remains optimal to select agents with the highest \(\ell\) values. Yet, if \(t \ge \ell\), the adversary can choose to make all selected agents byzantine, leading to a team of value zero. To provide meaningful guarantees, one hence needs to allow for randomization, in which case the expected value of the selected team needs to be maximized, assuming again that the adversary plays to minimize it. For this case, we provide linear-time randomized algorithms that maximize the expected value of the selected team.

A decade ago, Gerhard Woeginger posed an open problem that became well-known as "Woeginger’s Hiking Problem": Consider a group of \(n\) people that want to go hiking; everyone expresses preferences over the size of their hiking group in the form of an interval between \(1\) and \(n\). Is it possible to efficiently assign the \(n\) people to a set of hiking subgroups so that every person approves the size of their assigned subgroup? The problem is also known as efficiently deciding if an instance of an anonymous Hedonic Game with interval approval preferences admits a wonderful partition. We resolve the open problem in the affirmative by presenting an \(O(n^5)\) time algorithm for Woeginger’s Hiking Problem. Our solution is based on employing a dynamic programming approach for a specific rectangle stabbing problem from computational geometry. Moreover, we propose natural, more demanding extensions of the problem, e.g., maximizing the number of satisfied participants and variants with single-peaked preferences, and show that they are also efficiently solvable. Last but not least, we employ our solution to efficiently compute a partition that maximizes the egalitarian welfare for anonymous single-peaked Hedonic Games.

A group of \(n\) agents with numerical preferences for each other are to be assigned to the \(n\) seats of a dining table. We study two natural topologies: circular (cycle) tables and panel (path) tables. For a given seating arrangement, an agent’s utility is the sum of their preference values towards their (at most two) direct neighbors. An arrangement is envy-free if no agent strictly prefers someone else’s seat, and it is stable if no two agents strictly prefer each other’s seats. Recently, it was shown that for both paths and cycles it is NP-hard to decide whether an envy-free arrangement exists, even for symmetric binary preferences. In contrast, we show that, if agents come from a bounded number of classes, the problem is solvable in polynomial time for arbitrarily-valued possibly asymmetric preferences, including outputting an arrangement if possible. We also give simpler proofs of the previous hardness results if preferences are allowed to be asymmetric. For stability, it is known that deciding the existence of stable arrangements is NP-hard for both topologies, but only if sufficiently-many numerical values are allowed. As it turns out, even constructing unstable instances can be challenging in certain cases, e.g., binary values. We propose a complete characterization of the existence of stable arrangements based on the number of distinct values in the preference matrix and the number of agent classes. We also ask the same question for non-negative values and give an almost-complete characterization, the most interesting outstanding case being that of paths with two-valued non-negative preferences, for which we experimentally find that stable arrangements always exist and prove it under the additional constraint that agents can only swap seats when sitting at most two positions away. Similarly to envy-freeness, we also give a polynomial-time algorithm for determining a stable arrangement assuming a bounded number of classes. We moreover consider the swap dynamics and exhibit instances where they do not converge, despite a stable arrangement existing.