An algorithm for a super-stable roommates problem

In this paper, we describe an efficient algorithm that decides if a stable matching exists for a generalized stable roommates problem, where, instead of linear preferences, agents have partial preference orders on potential partners. Furthermore, we may forbid certain partnerships, that is, we are looking for a matching such that none of the matched pairs is forbidden, and yet, no blocking pair (forbidden or not) exists. To solve the above problem, we generalize the first algorithm for the ordinary stable roommates problem

An Algorithm for a Super-Stable Roommates Problem

In this paper we describe an efficient algorithm that decides if a stable matching exists for a generalized stable roommates problem, where, instead of linear preferences, agents have partial preference orders on potential partners. Furthermore, we may forbid certain partnerships, that is, we are looking for a matching such that none of the matched pairs is forbidden, and yet, no blocking pair (forbidden or not) exists. To solve the above problem, we generalize the first algorithm for the ordi- nary stable roommates problem

Stabil párosítások és általánosításaik = Stable matchings and its generalizations

A kutatási programunkban úgy érezzük, sikerült megvalósítani a kitűzött célokat. A csatolt publikációs listában szereplő 22 eredményünk többségét színvonalas nemzetközi folyóiratokban publikáltuk, vagy publikálni fogjuk. Számos nemzetközi konferencián vettünk részt, ahol ismertettük az eredményeinket és több kollégával szakmai együttműködést folytattunk. A kitűzött kutatási tervben az alábbi kutatási témák szerepeltek: blokkoló élek minimális száma (2 publikáció), stabil allokáció gráfokon (7 publikáció), Scarf lemma (1 publikáció), kooperatív játékelmélet (3 publikáció), gyakorlati alkalmazások (8 publikáció). Eredményeink ezeken kívül a stabil párosításoknak ill. azok általánosításainak létezésére ill. egyéb problémákban történő alkalmazásaira mutatnak rá. | We think that we succeeded to achieve our goal. Most of our 22 results in the attached list are published or will be published in high standard international journals. We participated several conferences, gave talks on these results and collaborated with colleagues. Our original research plan contains the following research topics: minimum number of blocking edges (2 publications), stable allocation on graphs (7 publications), Scarf's lemma (1 publication), cooperative game theory (3 publications), practical applications (8 publications). Beyond these, our results point out the existence of various generalizations of stable matchings and their applicability to other problems

Instability of Stable Matching: A Dynamic Trust Approach

Matching is of one of the essential problems in computer science and finding the stable matching is often difficult. For example, matching the medical students to hospitals where each student has its own preferences as well as each hospital has its own category in order to select the interns it can be difficult. Therefore there are many constraints that one needs to look at before solving matching problems. There are some algorithms which ensure that is possible to find a stable matching solution of this type of problems if any exists. In this paper, the instability of stable matching algorithm in dynamic environment has been demonstrated. The approach consists in mapping a specific stable matching problem (stable marriage problem) which has a stable solution into a dynamic environment where the instability is shown by using dynamic trust model

The Stable Marriage Problem with Ties and Restricted Edges

In the stable marriage problem, a set of men and a set of women are given, each of whom has a strictly ordered preference list over the acceptable agents in the opposite class. A matching is called stable if it is not blocked by any pair of agents, who mutually prefer each other to their respective partner. Ties in the preferences allow for three different definitions for a stable matching: weak, strong and super-stability. Besides this, acceptable pairs in the instance can be restricted in their ability of blocking a matching or being part of it, which again generates three categories of restrictions on acceptable pairs. Forced pairs must be in a stable matching, forbidden pairs must not appear in it, and lastly, free pairs cannot block any matching. Our computational complexity study targets the existence of a stable solution for each of the three stability definitions, in the presence of each of the three types of restricted pairs. We solve all cases that were still open. As a byproduct, we also derive that the maximum size weakly stable matching problem is hard even in very dense graphs, which may be of independent interest

Complementary cooperation, minimal winning coalitions, and power indices

We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). C-WMMG models a basic cooperation rule, the complementary cooperation rule, and can be taken as a sister model of the famous weighted majority game (WMG for short). In this paper, we concentrate on the two dimensional C-WMMG. An interesting property of this case is that there are at most $n+1$ minimal winning coalitions (MWC for short), and they can be enumerated in time $O(n\log n)$, where $n$ is the number of players. This property guarantees that the two dimensional C-WMMG is more handleable than WMG. In particular, we prove that the main power indices, i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel index, and the Deegan-Packel index, are all polynomially computable. To make a comparison with WMG, we know that it may have exponentially many MWCs, and none of the four power indices is polynomially computable (unless P=NP). Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. In WMG, this property is possessed by the Shapley-Shubik index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the Deegan-Packel index. Since our model fits very well the cooperation and competition in team sports, we hope that it can be potentially applied in measuring the values of players in team sports, say help people give more objective ranking of NBA players and select MVPs, and consequently bring new insights into contest theory and the more general field of sports economics. It may also provide some interesting enlightenments into the design of non-additive voting mechanisms. Last but not least, the threshold version of C-WMMG is a generalization of WMG, and natural variants of it are closely related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page