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Banach's match problem is a classic problem in probability attributed to Stefan Banach. Feller [ 1 ] says that the problem was inspired by a humorous reference to Banach's smoking habit in a speech honouring him by Hugo Steinhaus , but that it was not Banach who set the problem or provided an answer.
In a given instance of the stable-roommates problem (SRP), each of 2n participants ranks the others in strict order of preference. A matching is a set of n disjoint pairs of participants. A matching M in an instance of SRP is stable if there are no two participants x and y, each of whom prefers the other to their partner in M.
A stable matching always exists, and the algorithmic problem solved by the Gale–Shapley algorithm is to find one. [3] The stable matching problem has also been called the stable marriage problem, using a metaphor of marriage between men and women, and many sources describe the Gale–Shapley algorithm in terms of marriage proposals. However ...
Envy-free matching – a relaxation of stable matching for many-to-one matching problems; Rainbow matching for edge colored graphs; Stable matching polytope; Lattice of stable matchings; Secretary problem (also called marriage problem) – deciding when to stop to obtain the best reward in a sequence of options
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Fair random assignment (also called probabilistic one-sided matching) is a kind of a fair division problem. In an assignment problem (also called house-allocation problem or one-sided matching), there are m objects and they have to be allocated among n agents, such that each agent receives at most one object. Examples include the assignment of ...
By the birthday problem, the probability is close to 1 that at least two nodes in that subset are connected by an edge. In the same paper, the authors proposed a Boolean version of the problem, the Boolean Hidden Matching problem, and conjectured that the same quantum-classical gap holds for it as well. [1]
The matching problem can be generalized by assigning weights to edges in G and asking for a set M that produces a matching of maximum (minimum) total weight: this is the maximum weight matching problem. This problem can be solved by a combinatorial algorithm that uses the unweighted Edmonds's algorithm as a subroutine. [6]