The bottleneck distance is a natural measure of the distance between two finite point sets of equal cardinality. In this work, we consider the problem of indexing a set of @math planar point sets (of varying sizes) to create a database @math that supports nearest bottleneck distance queries: given a query point set @math of size @math , the point sets @math that are closest in terms of bottleneck distance are returned. Without loss of generality, we assume that all point sets belong to the unit box @math in the plane. The main contribution of this work is a trie -based data structure that is space efficient and supports @math -approximate nearest bottleneck queries in @math time, where @math is the minimum bottleneck distance from @math to any point set in @math . A direct consequence, of independent interest, is a simple @math time algorithm to @math -approximate @math , for any two point sets @math and @math . Finally, the querying algorithm proposed is easily adapted to support nearest subset and superset queries.
Approximation results are known for general bipartite matching in metric spaces; in @cite_2 the authors show that for any @math , there is an algorithm that computes a @math -approximate matching, where @math in @math time. A variation on minimum-distance bottleneck matching, with the additional constraint that the matched edges cannot cross, was recently shown to be NP-hard to approximate within a factor of less than @math @cite_7 .
@cite_7 @cite_2
2079168081 1629950647 2951871496 2044105411
Let G = G(A∪B,A×B), with |A| = |B| = n, be a weighted bipartite graph, and let d(·,·) be the cost function on the edges. Let w(M) denote the weight of a matching in G, and M* a minimum-cost perfect matching in G. We call a perfect matching M c-approximate, for c ≥ 1, if w(M) ≤ c · w(M*). We present three approximation algorithms for computing minimum-cost perfect matchings in G. First, we consider the case when d(·,·) is a metric. For any Δ > 0, we present an algorithm that, in O(n2+Δ log n log2(1 Δ)) time, computes a O(1 Δα)-approximate matching of G, where α = log3 2 ≈ 0.631. Next, we assume the existence of a dynamic data structure for answering approximate nearest neighbor (ANN) queries under d(··). Given two parameters e, d e) is the query and update time of an (e 2)-ANN data structure. Finally, we present an algorithm that works even if d(·,·) is not a metric but admits an ANN data structure for d(·,·). In particular, we present an algorithm that computes, in O(e---1n3 2τ (n, e) log4(n e) log Δ) time, a (1 + e)-approximate matching of A and B; here Δ is the ratio of the largest to the smallest-cost edge in G, and τ (n, e) is the query and update time of an (e c)-ANN data structure for some constant c > 1. We show that our results lead to faster matching algorithms for many geometric settings. In this paper, we study the non-bipartite maximum matching problem in the semi-streaming model. The maximum matching problem in the semi-streaming model has received a significant amount of attention lately. While the problem has been somewhat well solved for bipartite graphs, the known algorithms for non-bipartite graphs use @math passes or @math time to compute a @math approximation. In this paper we provide the first FPTAS (polynomial in @math ) for the problem which is efficient in both the running time and the number of passes. We also show that we can estimate the size of the matching in @math passes using slightly superlinear space. To achieve both results, we use the structural properties of the matching polytope such as the laminarity of the tight sets and total dual integrality. The algorithms are iterative, and are based on the fractional packing and covering framework. However the formulations herein require exponentially many variables or constraints. We use laminarity, metric embeddings and graph sparsification to reduce the space required by the algorithms in between and across the iterations. This is the first use of these ideas in the semi-streaming model to solve a combinatorial optimization problem. In this paper we consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. We are given an undirected graph in which every edge is assigned a probability of existence and a positive profit, and each node is assigned a positive integer called timeout. We know whether an edge exists or not only after probing it. On this random graph we are executing a process, which one-by-one probes the edges and gradually constructs a matching. The process is constrained in two ways: once an edge is taken it cannot be removed from the matching, and the timeout of node @math upper-bounds the number of edges incident to @math that can be probed. The goal is to maximize the expected profit of the constructed matching. For this problem (Algorithmica 2012) provided a @math -approximation algorithm for bipartite graphs, and a @math -approximation for general graphs. In this work we improve the approximation factors to @math and @math , respectively. We also consider an online version of the bipartite case, where one side of the partition arrives node by node, and each time a node @math arrives we have to decide which edges incident to @math we want to probe, and in which order. Here we present a @math -approximation, improving on the @math -approximation of The main technical ingredient in our result is a novel way of probing edges according to a random but non-uniform permutation. Patching this method with an algorithm that works best for large probability edges (plus some additional ideas) leads to our improved approximation factors. Let G=(U, V, E) be a bipartite graph with |U|=|V|=n. The factor size of G, f, is the maximum number of edge disjoint perfect matchings in G. We characterize the complexity of counting the number of perfect matchings in classes of graphs parameterized by factor size. We describe the simple algorithm, which is an approximation algorithm for the permanent that is a natural simplification of the algorithm suggested by Broder (1986) and analyzed by Jerrum and Sinclair (1988a, b). Compared to the algorithm by Jerrum and Sinclair (1988a, b), the simple algorithm achieves a polynomial speed up in the running time to compute the permanent. A combinatorial lemma is used to prove that the simple algorithm runs in time nO(n f). Thus: (1) for all constants α>0, the simple algorithm runs in polynomial time for graphs with factor size at least αn; (2) for some constant c, the simple algorithm is the fastest known approximation for graphs with factor size at least c log n. (Compare with the approximation algorithms described in Karmarkar (1988).) We prove the following complementary hardness results. For functions f such that 3⩽f(n)⩽n−3, the exact counting problem for f(n)-regular bipartite graphs is #P-complete. For and e>0, for any function f such that 3⩽f(n)⩽n1−e, approximate counting for f(n)-regular bipartite graphs is as hard as approximate counting for all bipartite graphs. An announcement of these results appears in Dagum (1988).