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
2289848075 2079168081
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. 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. A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Many classical optimization problems, such as maximum spanning tree and forest, shortest path, maximum weight (perfect) matching, maximum weight independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. Not much is known however about the case of two or more budgets: filling this gap, at least partially, is the main goal of this paper. In more detail, we obtain the following main results: Using iterative rounding for the first time in multi-objective optimization, we obtain multi-criteria PTASs (which slightly violate the budget constraints) for spanning tree, matroid basis, and bipartite matching with @math k = O ( 1 ) budget constraints. We present a simple mechanism to transform multi-criteria approximation schemes into pure approximation schemes for problems whose feasible solutions define an independence system. This gives improved algorithms for several problems. In particular, this mechanism can be applied to the above bipartite matching algorithm, hence obtaining a pure PTAS. We show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for @math k -budgeted matroid independent set. We present a deterministic approximation scheme for @math k -budgeted matching (in general graphs), where @math k = O ( 1 ) . Interestingly, to show that our procedure works, we rely on a non-constructive result by Stromquist and Woodall, which is based on the Ham Sandwich Theorem. In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input using O(n) @math n @math 1-1 e @math O(n) @math 1-1 e @math O(n) @math 1-1 e @math O(n) @math 1-e^ -k k^ k-1 (k-1)!=1- 2 k +o(1 k) @math k$ passes in the vertex arrival model using linear space, improving upon previously best known convergence.