Matching with Our Eyes Closed

Abstract

Abstract-Motivated by an application in kidney exchange, we study the following query-commit problem: we are given the set of vertices of a non-bipartite graph G. The set of edges in this graph are not known ahead of time. We can query any pair of vertices to determine if they are adjacent. If the queried edge exists, we are committed to match the two endpoints. Our objective is to maximize the size of the matching.This restriction in the amount of information available to the algorithm constraints us to implement myopic, greedy-like algorithms. A simple deterministic greedy algorithm achieves a factor 1/2 which is tight for deterministic algorithms. An important open question in this direction is to give a randomized greedy algorithm that has a significantly better approximation factor. This question was first asked almost 20 years ago by Dyer and Frieze [9] where they showed that a natural randomized strategy of picking edges uniformly at random doesn't help and has an approximation factor of 1/2 + o(1). They left it as an open question to devise a better randomized greedy algorithm. In subsequent work, Aronson, Dyer, Frieze, and Suen [2] gave a different randomized greedy algorithm and showed that it attains a factor 0.5 + ǫ where ǫ is 0.0000025.In this paper we propose and analyze a new randomized greedy algorithm for finding a large matching in a general graph and use it to solve the query commit problem mentioned above. We show that our algorithm attains a factor of at least 0.56, a significant improvement over 0.50000025. We also show that no randomized algorithm can have an approximation factor better than 0.7916 for the query commit problem. For another large and interesting class of randomized algorithms that we call vertex-iterative algorithms, we show that no vertexiterative algorithm can have an approximation factor better than 0.75.

Ignorance is Almost Bliss

“…A popular variant is the querycommit problem, where the algorithm is forced to add any queried edge to the matching if the edge is found to exist. Goel and Tripathi [2012] establish an upper bound of 0.7916 for graphs in which no information is available about the edges, while Costello et al [2012] establish a lower bound of 0.573 and an upper bound of 0.898 for graphs in which each edge e exists with a given probability p e . Similarly to our work, these approximation ratios are with respect to the omniscient optimum, but the informational disadvantage of the algorithm stems purely from the query-commit restriction.…”

Randomized Greedy Algorithms for the Maximum Matching Problem with New Analysis

“…While MRG has been studied extensively, RANKING (for general graphs) is a recent invention of Jukna and Schnitger (also, independently in [16], [17]), inspired by the ideas in [11].…”

“…The lower bound method of Aronson, Dyer, Frieze and Suen breaks down, because RANKING uses only a fraction of the randomness that MRG does, which makes it harder to apply any independence argument. Very recently we have learned that Pushkar Tripathi, independently from us, has invented and studied RANKING, and in his thesis [16] establishes (referring to joint work [17]):…”

Подвижность Электронов В Инверсионных Слоях Полностью Обедняемых Пленок Кремний-На-Изоляторе

“…Подвижность носителей заряда, ограниченная рассея-нием на микрорельефе ГР, не зависит от температуры, µ s r = f (T ) = const [20][21][22]. Подвижность, ограниченная рассеянием на поверхностных фононах, согласно модели Ломбарди [5,7], обратно пропорциональна температуре, µ s ph (T ) = A + bT −1 , где коэффициенты a и b зависят от напряженности эффективного поля в канале (a ∝ E области 3 выражение (3) можно записать в виде…”