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. We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass @math -approximation streaming algorithm can be easily obtained with space @math , where @math denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space @math is granted, for every @math . Specifically, for every @math , we show that in the one-pass turnstile streaming model, in order to compute a @math -approximation, space @math is required for constant error randomized algorithms, and, up to logarithmic factors, space @math is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest. We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of @math , a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the @math bound.Our main result is a @math -approximation online algorithm for stochastic bipartite matching, breaking this @math barrier. Furthermore, we show that no online algorithm can produce a @math approximation for an arbitrarily small @math for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-) matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application. We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems. Our technique produces approximation algorithms that run in @math time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of @math time compares favorably with the best strongly polynomial exact algorithms running in @math time for dense graphs. A similar result is obtained for the 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413--420]. We study approximate string matching in connection with two string distance functions that are computable in linear time. The first function is based on the so-called @math -grams. An algorithm is given for the associated string matching problem that finds the locally best approximate occurences of pattern @math , @math , in text @math , @math , in time @math . The occurences with distance @math can be found in time @math . The other distance function is based on finding maximal common substrings and allows a form of approximate string matching in time @math . Both distances give a lower bound for the edit distance (in the unit cost model), which leads to fast hybrid algorithms for the edit distance based string matching. We consider distributed algorithms for approximate maximum matching on general graphs. Our main result is a randomized @math -approximation distributed algorithm for maximum weighted matching, whose running time is @math for any constant @math , where @math is the number of nodes in the graph. This is, to the best of our knowledge, the first log-time distributed algorithm that achieves constant approximation for maximum weighted matching on general graphs. In addition, we consider the dynamic case, where nodes are inserted and deleted one at a time. For unweighted dynamic graphs, we give a distributed algorithm that maintains a @math -approximation in @math time for each node insertion or deletion for any constant @math . For weighted dynamic graphs we give a constant-factor approximation distributed algorithm that runs in constant time for each insertion or deletion. We consider the @math -Center problem and some generalizations. For @math -Center a set of @math center vertices needs to be found in a graph @math with edge lengths, such that the distance from any vertex of @math to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter @math and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameter @math , but implies that such an algorithm must have at least doubly exponential running time in @math if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of @math by combining the parameters @math and @math : we develop a fixed-parameter @math -approximation with running time @math . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter @math -approximations for only the parameter @math . We also provide similar fixed-parameter approximations for the weighted @math -Center and @math -Partition problems, which generalize @math -Center. In the distributed all-pairs shortest paths problem (APSP), every node in the weighted undirected distributed network (the CONGEST model) needs to know the distance from every other node using least number of communication rounds (typically called time complexity ). The problem admits @math -approximation @math -time algorithm and a nearly-tight @math lower bound [Nanongkai, STOC'14; Lenzen and Patt-Shamir PODC'15] @math , @math and @math hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios LenzenP_podc13,HolzerW12,PelegRT12,Nanongkai-STOC14 . . For the exact case, Elkin [STOC'17] presented an @math time bound, which was later improved to @math [Huang, Nanongkai, Saranurak FOCS'17]. It was shown that any super-linear lower bound (in @math ) requires a new technique [Censor-Hillel, Khoury, Paz, DISC'17], but otherwise it remained widely open whether there exists a @math -time algorithm for the exact case, which would match the best possible approximation algorithm. This paper resolves this question positively: we present a randomized (Las Vegas) @math -time algorithm, matching the lower bound up to polylogarithmic factors. Like the previous @math bound, our result works for directed graphs with zero (and even negative) edge weights. In addition to the improved running time, our algorithm works in a more general setting than that required by the previous @math bound; in our setting (i) the communication is only along edge directions (as opposed to bidirectional), and (ii) edge weights are arbitrary (as opposed to integers in 1, 2, ... poly(n) ). ... We study constant-time approximation algorithms for bounded-degree graphs, which run in time independent of the number of vertices @math . We present an algorithm that decides whether a vertex is contained in a some fixed maximal independent set with expected query complexity @math , where @math is the degree bound. Using this algorithm, we show constant-time approximation algorithms with certain multiplicative error and additive error @math for many other problems, e.g., the maximum matching problem, the minimum vertex cover problem, and the minimum set cover problem, that run exponentially faster than existing algorithms with respect to @math and @math . Our approximation algorithm for the maximum matching problem can be transformed to a two-sided error tester for the property of having a perfect matching. On the contrary, we show that every one-sided error tester for the property requires at least @math queries. We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products @math based on a set of customer samples @math . We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor @math for some constant @math , unless @math , thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within @math , @math being an upper bound on the number of nonzero values per customer, and @math under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas. Introducing a slightly more general problem definition in which customers are given as an explicit probability distribution, we obtain inapproximability within @math assuming @math . These results apply to single-minded pricing as well. For the max-buying model a polynomial-time approximation scheme exists if a price ladder is given. We give a matching lower bound by proving strong NP-hardness. Assuming limited product supply, we analyze a generic local search algorithm and prove that it is 2-approximate. Finally, we discuss implications for the rank-buying model. The Euclidean @math -means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of @math points in Euclidean space @math , and the goal is to choose @math centers in @math so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general @math and a @math -approximation which runs in time @math . At the other extreme, the only known computational complexity result for this problem is NP-hardness [ADHP'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in @math can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all @math . In this paper we provide the first hardness of approximation for the Euclidean @math -means problem. Concretely, we show that there exists a constant @math such that it is NP-hard to approximate the @math -means objective to within a factor of @math . We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform @math , a known hard vertex cover instance, by taking a graph product with a suitably chosen graph @math , and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest. In an instance of the stable marriage problem with ties and incomplete preference lists, stable matchings can have different sizes. It is APX-hard to compute a maximum cardinality stable matching, but there have recently been proposed polynomial-time approximation algorithms, with constant performance guarantees for both the general version of this problem, and for several special cases. Our contribution is to describe a @math -approximation algorithm for the general version of this problem, improving upon the recent @math -approximation algorithm of Kiraly. Interest in such algorithms arises because of the problem's application to centralized matching schemes, the best known of which involve the assignment of graduating medical students to hospitals in various countries. We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching Mis popular if there is no matching Mi¾? such that more people prefer Mi¾? to Mthan the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied in [2]. If there is no popular matching, a reasonable substitute is a matching whose unpopularityis bounded. We consider two measures of unpopularity - unpopularity factordenoted by u(M) and unpopularity margindenoted by g(M). McCutchen recently showed that computing a matching Mwith the minimum value of u(M) or g(M) is NP-hard, and that if Gdoes not admit a popular matching, then we have u(M) i¾? 2 for all matchings Min G. Here we show that a matching Mthat achieves u(M) = 2 can be computed in @math time (where mis the number of edges in Gand nis the number of nodes) provided a certain graph Hadmits a matching that matches all people. We also describe a sequence of graphs: H= H 2 , H 3 ,...,H k such that if H k admits a matching that matches all people, then we can compute in @math time a matching Msuch that u(M) ≤ ki¾? 1 and @math . Simulation results suggest that our algorithm finds a matching with low unpopularity. We present a simple distributed @math -approximation algorithm for maximum weight independent set (MaxIS) in the @math model which completes in @math rounds, where @math is the maximum degree, @math is the number of rounds needed to compute a maximal independent set (MIS) on @math , and @math is the maximum weight of a node. Whether our algorithm is randomized or deterministic depends on the MIS algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in @math rounds, where @math is the number of nodes. We also present a deterministic @math -round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a @math -approximation for maximum weight matching without incurring any additional round penalty in the @math model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of for which we describe a mechanism that allows the simulation to run in the @math model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ( @math ) allows us to devise a distributed algorithm requiring @math rounds for any constant @math . For the unweighted case, we can even obtain a @math -approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on @math . We consider the maximum independent set problem on graphs with maximum degree @math . We show that the integrality gap of the Lov 'asz @math -function based SDP is @math . This improves on the previous best result of @math , and almost matches the integrality gap of @math recently shown for stronger SDPs, namely those obtained using poly- @math levels of the @math semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of @math -free graphs for large values of @math . We also show how to obtain an algorithmic version of the above-mentioned @math -based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of @math matches the best unique-games-based hardness result up to lower-order poly- @math factors. We consider the k-Center problem and some generalizations. For k-Center a set of kcenter vertices needs to be found in a graph G with edge lengths, such that the distance from any vertex ofi¾?G to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [ICALP 2011]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter k, and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameteri¾?h, but implies that such an algorithm must have at least doubly exponential running time in h if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor ofi¾?2 by combining the parameters k andi¾?h: we develop a fixed-parameter 3 2-approximation with running time @math . We also provide similar fixed-parameter approximations for the weightedk-Center and @math -Partition problems, which generalize k-Center. We give the first improvement to the space approximation trade-off of distance oracles since the seminal result of Thorup and Zwick [STOC'01]. For unweighted graphs, our distance oracle has size @math and, when queried about vertices at distance @math , returns a path of length @math . For weighted graphs with @math edges, our distance oracle has size @math and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space @math . For unweighted graphs, this implies a @math space lower bound to achieve approximation @math .