TY - JOUR A1 - Cseh, Agnes A1 - Heeger, Klaus T1 - The stable marriage problem with ties and restricted edges JF - Discrete optimization N2 - 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. KW - stable matchings KW - restricted edges KW - complexity Y1 - 2020 U6 - https://doi.org/10.1016/j.disopt.2020.100571 SN - 1572-5286 SN - 1873-636X VL - 36 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Cseh, Ágnes A1 - Fleiner, Tamas T1 - The complexity of cake cutting with unequal shares JF - ACM transactions on algorithms : TALG N2 - An unceasing problem of our prevailing society is the fair division of goods. The problem of proportional cake cutting focuses on dividing a heterogeneous and divisible resource, the cake, among n players who value pieces according to their own measure function. The goal is to assign each player a not necessarily connected part of the cake that the player evaluates at least as much as her proportional share.
In this article, we investigate the problem of proportional division with unequal shares, where each player is entitled to receive a predetermined portion of the cake. Our main contribution is threefold. First we present a protocol for integer demands, which delivers a proportional solution in fewer queries than all known protocols. By giving a matching lower bound, we then show that our protocol is asymptotically the fastest possible. Finally, we turn to irrational demands and solve the proportional cake cutting problem by reducing it to the same problem with integer demands only. All results remain valid in a highly general cake cutting model, which can be of independent interest. KW - Cake cutting KW - fair division KW - unequal shares KW - proportional division Y1 - 2020 U6 - https://doi.org/10.1145/3380742 SN - 1549-6325 SN - 1549-6333 VL - 16 IS - 3 PB - Association for Computing Machinery CY - New York ER - TY - JOUR A1 - Cseh, Ágnes A1 - Juhos, Attila T1 - Pairwise preferences in the stable marriage problem JF - ACM Transactions on Economics and Computation / Association for Computing Machinery N2 - We study the classical, two-sided stable marriage problem under pairwise preferences. In the most general setting, agents are allowed to express their preferences as comparisons of any two of their edges, and they also have the right to declare a draw or even withdraw from such a comparison. This freedom is then gradually restricted as we specify six stages of orderedness in the preferences, ending with the classical case of strictly ordered lists. We study all cases occurring when combining the three known notions of stability-weak, strong, and super-stability-under the assumption that each side of the bipartite market obtains one of the six degrees of orderedness. By designing three polynomial algorithms and two NP-completeness proofs, we determine the complexity of all cases not yet known and thus give an exact boundary in terms of preference structure between tractable and intractable cases. KW - Stable marriage KW - intransitivity KW - acyclic preferences KW - poset KW - weakly KW - stable matching KW - strongly stable matching KW - super stable matching Y1 - 2021 U6 - https://doi.org/10.1145/3434427 SN - 2167-8375 SN - 2167-8383 VL - 9 IS - 1 PB - Association for Computing Machinery CY - New York ER - TY - JOUR A1 - Cseh, Ágnes A1 - Kavitha, Telikepalli T1 - Popular matchings in complete graphs JF - Algorithmica : an international journal in computer science N2 - Our input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching M ': here each vertex casts a vote for the matching in {M,M '} in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes NP-complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and NP-complete when n has the other parity. KW - Popular matching KW - Complexity KW - Stable matching Y1 - 2021 U6 - https://doi.org/10.1007/s00453-020-00791-7 SN - 0178-4617 SN - 1432-0541 VL - 83 IS - 5 SP - 1493 EP - 1523 PB - Springer CY - New York ER - TY - JOUR A1 - Cseh, Agnes A1 - Faenza, Yuri A1 - Kavitha, Telikepalli A1 - Powers, Vladlena T1 - Understanding popular matchings via stable matchings JF - SIAM journal on discrete mathematics N2 - An instance of the marriage problem is given by a graph G = (A boolean OR B, E), together with, for each vertex of G, a strict preference order over its neighbors. A matching M of G is popular in the marriage instance if M does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is a min-size popular matching; another subclass of popular matchings that always exists and can be easily computed is the set of dominant matchings. A popular matching M is dominant if M wins the head-to-head election against any larger matching. Thus, every dominant matching is a max-size popular matching, and it is known that the set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. Results from the literature seem to suggest that stable and dominant matchings behave, from a complexity theory point of view, in a very similar manner within the class of popular matchings. The goal of this paper is to show that there are instead differences in the tractability of stable and dominant matchings and to investigate further their importance for popular matchings. First, we show that it is easy to check if all popular matchings are also stable; however, it is co-NP hard to check if all popular matchings are also dominant. Second, we show how some new and recent hardness results on popular matching problems can be deduced from the NP-hardness of certain problems on stable matchings, also studied in this paper, thus showing that stable matchings can be employed to show not only positive results on popular matchings (as is known) but also most negative ones. Problems for which we show new hardness results include finding a min-size (resp., max-size) popular matching that is not stable (resp., dominant). A known result for which we give a new and simple proof is the NP-hardness of finding a popular matching when G is nonbipartite. KW - popular matching KW - stable matching KW - complexity KW - dominant matching Y1 - 2022 U6 - https://doi.org/10.1137/19M124770X SN - 0895-4801 SN - 1095-7146 VL - 36 IS - 1 SP - 188 EP - 213 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER -