Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GREEDYMATCHING. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GREEDYMATCHING is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GREEDYMATCHING is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider three natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and computational tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GREEDYMATCHING on a special class of weighted graphs, called bushgraphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is ρ, then for every ε > 0 the randomized MRG algorithm of (Aronson et al. 1995) gives a (ρ − ε)-approximation for the maximum cardinality matching. We conjecture that a tightbound for ρ is 2/3; we prove our conjecture true for four subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for ρ) is a long-standing open problem (Poloczek and Szegedy 2012). This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem.
Stability is a central concept in exchange-based mechanismdesign. It imposes a fundamental requirement that no subsetof agents could beneficially deviate from the outcome pre-scribed by the mechanism. However, deployment of stabilityin an exchange mechanism presents at least two challenges.First, it reduces social welfare and sometimes prevents themechanism from producing a solution. Second, it might incurcomputational cost to clear the mechanism.In this paper, we propose an alternative notion of stability,coined internal stability, under which we analyze the socialwelfare bounds and computational complexity. Our contribu-tions are as follows: for both pairwise matchings and limited-length exchanges, for both unweighted and weighted graph-s, (1) we prove desirable tight social welfare bounds; (2) weanalyze the computational complexity for clearing the match-ings and exchanges. Extensive experiments on the kidney ex-change domain demonstrate that the optimal welfare underinternal stability is very close to the unconstrained optimal.
Chan, Pak Hay (The Chinese University of Hong Kong) | Huang, Xin (The Chinese University of Hong Kong) | Liu, Zhengyang (Shanghai Jiao Tong University) | Zhang, Chihao (Shanghai Jiao Tong University) | Zhang, Shengyu (The Chinese University of Hong Kong)
We introduce a roommate market model, in which 2n people need to be assigned to n rooms, with two people in each room. Each person has a valuation to each room, as well as a valuation to each of other people as a roommate. Each room has a rent shared by the two people living in the room, and we need to decide who live together in which room and how much each should pay. Various solution concepts on stability and envy-freeness are proposed, with their existence studied and the computational complexity of the corresponding search problems analyzed. In particular, we show that maximizing the social welfare is NP-hard, and we give a polynomial time algorithm that achieves at least 2/3 of the maximum social welfare. Finally, we demonstrate a pricing scheme that can achieve envy-freeness for each room.
Stable Marriage (SM) is a well-known matching problem, where the aim is to match a set of men and women. The resulting matching must satisfy two properties: there is no unassigned person and there are no other assignments where two people of opposite gender prefer each other to their current assignments. We propose a new version of SM called as Robust Stable Marriage (RSM) by combining stability and robustness. We define robustness by introducing (a,b)-supermatches, which has been inspired by (a,b)-supermodels. An (a,b)-supermatch is a stable matching, where if at most a pairs want to break up, it is possible to find another stable matching by breaking at most b other pairs.
It is well-known that the Gale-Shapley algorithm is not truthful for all agents. Previous studies in this category concentrate on manipulations using incomplete preference lists by a single woman and by the set of all women. Little is known about manipulations by a subset of women. In this paper, we consider manipulations by any subset of women with arbitrary preferences. We show that a strong Nash equilibrium of the induced manipulation game always exists among the manipulators and the equilibrium outcome is unique and Pareto-dominant. In addition, the set of matchings achievable by manipulations has a lattice structure. We also examine the super-strong Nash equilibrium in the end.