The Stable Roommates problem with short lists

We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists (sri) that are degree constrained, i.e., preference lists are of bounded length. The first variant, egald-sri, involves finding an egalitarian stable matching in solvable instances of sri with preference lists of length at most d. We show that this problem is NP-hard even if d = 3. On the positive side we give a 2d+372d+37-approximation algorithm for d ∈{3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called d-srti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-srti admits a stable matching is NP-complete even if d = 3. We also consider the “most stable” version of this problem and prove a strong inapproximability bound for the d = 3 case. However for d = 2 we show that the latter problem can be solved in polynomial time

On absolutely and simply popular rankings

Van Zuylen et al. introduced the notion of a popular ranking in a voting context, where each voter submits a strictly-ordered list of all candidates. A popular ranking $\pi$ of the candidates is at least as good as any other ranking $\sigma$ in the following sense: if we compare $\pi$ to $\sigma$, at least half of all voters will always weakly prefer~$\pi$. Whether a voter prefers one ranking to another is calculated based on the Kendall distance. A more traditional definition of popularity -- as applied to popular matchings, a well-established topic in computational social choice -- is stricter, because it requires at least half of the voters \emph{who are not indifferent between $\pi$ and $\sigma$} to prefer~$\pi$. In this paper, we derive structural and algorithmic results in both settings, also improving upon the results by van Zuylen et al. We also point out strong connections to the famous open problem of finding a Kemeny consensus with 3 voters.Comment: full version of the AAMAS 2021 extended abstract 'On weakly and strongly popular rankings

Organizing time banks: Lessons from matching markets

A time bank is a group of people that set up a common platform to trade services among themselves. There are several well-known problems associated with this type of time banking, e.g., high overhead costs and difficulties to identify feasible trades. This paper constructs a non-manipulable mechanism that selects an individually rational and time-balanced allocation which maximizes exchanges among the members of the time bank (and those allocations are efficient). The mechanism works on a domain of preferences where agents classify services as unacceptable and acceptable (and for those services agents have specific upper quotas representing their maximum needs)

Organizing Time Exchanges: Lessons from Matching Markets

This paper considers time exchanges via a common platform (e.g., markets for exchanging time units, positions at education institutions, and tuition waivers). There are several problems associated with such markets, e.g., imbalanced outcomes, coordination problems, and inefficiencies. We model time exchanges as matching markets and construct a non-manipulable mechanism that selects an individually rational and balanced allocation which maximizes exchanges among the participating agents (and those allocations are efficient). This mechanism works on a preference domain whereby agents classify the goods provided by other participating agents as either unacceptable or acceptable, and for goods classified as acceptable agents have specific upper quotas representing their maximum needs

On Weakly and Strongly Popular Rankings

Van Zuylen et al. introduced the notion of a popular ranking in a voting context, where each voter submits a strictly-ordered list of all candidates. A popular ranking pi of the candidates is at least as good as any other ranking sigma in the following sense: if we compare pi to sigma, at least half of all voters will always weakly prefer pi. Whether a voter prefers one ranking to another is calculated based on the Kendall distance. A more traditional definition of popularity---as applied to popular matchings, a well-established topic in computational social choice---is stricter, because it requires at least half of the voters who are not indifferent between pi and sigma to prefer pi. In this paper, we derive structural and algorithmic results in both settings, also improving upon the results by van Zylen et al. We also point out connections to the famous open problem of finding a Kemeny consensus with 3 voters

Komplexität und Algorithmen in Matchingproblemen mit Präferenzen

In dieser Arbeit fokussieren wir uns auf Matching- und Flussprobleme auf Instanzen mit Präferenzen. Es werden Märkte mit Hilfe von kombinatorischer Optimierung durch Graphen modelliert. Während Hersteller, Händler und Verbraucher durch Knoten repräsentiert werden, sind die möglichen Geschäfte zwischen ihnen als Kanten veranschaulicht. Abhängig von der Struktur des Marktes wird einen Warenfluss oder eine Händlerpaarung gesucht. Das grundlegende Prinzip ist immer das gleiche: Stabilität. Eine Marktsituation ist stabil, wenn kein Paar von Händlern existiert, die die Situation bei gegenseitiger Zustimmung ändern möchten. Abhängig von der Komplexität des Problems unterscheiden wir drei Instanzen: Matching-, Allokations- und Flussinstanz. Die Spätere kann immer als Verallgemeinerung der Frühere betrachtet werden. Eine stabile Lösung zu finden ist in allen drei Problemen bewältigbar. Kapitel 1: Grundlagen in stabilen Matchings. Das Konzept Stabilität wird eingeführt und die wichtigsten Resultate aus der Literatur werden erwähnt. Wir geben Beispiele für Anwendungen. Kapitel 2: Stabile Matchings mit beschränkten Kanten. In diesem Kapitel analysieren wir Matchingsprobleme auf bipartiten und nichtbipartiten Graphen, die beschränkte Kanten enthalten. Eine beschränkte Kante ist entweder erzwungen oder verboten. Wir betrachten zwei Approximationskonzepte und bieten eine komplette Analyse aller möglichen Fällen an. Kapitel 3: Weitere Komplexitätsresultate für stabilen Matchings. Hier werden zwei weitere Probleme auf Matchingsinstanzen diskutiert. In dem ersten suchen wir ein Matching mit freien Kanten, das maximale Kardinalität hat. In der zweiten Hälfte des Kapitels wird ein stabiles Matching in nichtbipartiten Graphen mit Gleichstand in Präferenzlisten gesucht und NP-Härte gezeigt. Kapitel 4: Pfade zur stabilen Allokationen. Das stabile Allokationsproblem ist eine kapazitierte Variante des stabilen Matchingsproblems. In diesem Kapitel stellen wir die Frage, ob random und deterministische Prozesse auf unkontrollierten Märkten terminieren. Laufzeiten der besser und best response Strategien sind analysiert. Kapitel 5: Nicht teilbare stabile Allokationen. Eine natürliche Erweiterung des stabilen Allokationsproblems ist die ganzzahlige Variante: Hier sucht man Allokationen, in den Elemente nicht geteilt werden. Wir zeigen dass das Problem in Polynomialzeit bewältigbar ist und präsentieren einen Rundungsalgorithmus. Kapitel 6: Stabile Flüsse. Wie Matchingsprobleme in allgemeinen, stabile Matchings können auch auf Flussinstanzen verallgemeinert werden. Wir präsentieren einen polynomialen Algorithmus, eine Methode für stabile Flüsse mit beschränkten Kanten und einen Beweis für die Härte integraler multicommodity Flüssen. Kapitel 7: Populäre Matchings. In dem letzten Kapitel wird eine Alternative zur stabilen Matchings diskutiert. Die Komplexität von zwei Problemen in dem Thema wird betrachtet.In this thesis we focus on matching markets under preferences - each market participant expresses their preferences as an ordered list of possible scenarios. Our task is to find a matching that is optimal with respect to these preferences. The most common notion of optimality is stability. A matching is stable if no two unmatched agents prefer each other to their respective partners. We discuss various problems in stable matchings from the algorithmic point of view, either presenting an efficient algorithm for solving them or proving hardness. Chapter 1: Basic notions in stable matchings. We introduce the concept of stable matchings formally and present some of the most important theorems related to it, including the Gale-Shapley algorithm. Real-life applications are also discussed briefly. Chapter 2: Stable marriage and roommates problems with restricted edges. We start with classical one-to-one matchings in bipartite and non-bipartite graphs and investigate the problem of stable matchings when forced and forbidden edges are present. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs. This chapter is a comprehensive study of complexity and approximability results. Chapter 3: Other complexity results for stable matchings. In this chapter we investigate two problems in the stable matching setting: the maximum cardinality stable matching problem and a degree constrained version of the stable roommates problem with ties. In both cases, we show intractability. Chapter 4: Paths to stable allocations. We introduce the stable allocation problem, a capacitated variant of the stable matching problem. We analyze both better and best response dynamics in uncoordinated markets from an algorithmic point of view and discuss deterministic and random procedures as well. Chapter 5: Unsplittable stable allocation problems. In this chapter, we study a natural unsplittable variant of the stable allocation problem. Our main result is to show that the problem is solvable in polynomial time. We also elaborate on relaxed solutions and rounding methods. Chapter 6: Stable flows. As most matching problems, stable matchings also can be extended to network flows. In this chapter we present the polynomial version of the Gale-Shapley algorithm for stable flows, solve the stable flows with forced and forbidden edges problem and establish the complexity of the integral stable multicommodity flows problem. Chapter 7: Popular matchings. In the last chapter we discuss an alternative optimality notion to stability, popularity. We investigate two problems under this setting