3,500 research outputs found

    New Iterative Algorithms for Weighted Matching

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    Matching is an important combinatorial problem with a number ofpractical applications. Even though there exist polynomial time solutionsto most matching problems, there are settings where these are too slow.This has led to the development of several fast approximation algorithmsthat in practice compute matchings very close to the optimal.The current paper introduces a new deterministic approximationalgorithm named G 3 , for weighted matching. The algorithm is based ontwo main ideas, the first is to compute heavy subgraphs of the originalgraph on which we can compute optimal matchings. The second idea isto repeatedly merge these matchings into new matchings of even higherweight than the original ones. Both of these steps are achieved usingdynamic programming in linear or close to linear time.We compare G 3 with the randomized algorithm GPA+ROMA whichis the best known algorithm for this problem. Experiments on alarge collection of graphs show that G 3 is substantially faster thanGPA+ROMA while computing matchings of comparable weight

    Risk-Averse Matchings over Uncertain Graph Databases

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    A large number of applications such as querying sensor networks, and analyzing protein-protein interaction (PPI) networks, rely on mining uncertain graph and hypergraph databases. In this work we study the following problem: given an uncertain, weighted (hyper)graph, how can we efficiently find a (hyper)matching with high expected reward, and low risk? This problem naturally arises in the context of several important applications, such as online dating, kidney exchanges, and team formation. We introduce a novel formulation for finding matchings with maximum expected reward and bounded risk under a general model of uncertain weighted (hyper)graphs that we introduce in this work. Our model generalizes probabilistic models used in prior work, and captures both continuous and discrete probability distributions, thus allowing to handle privacy related applications that inject appropriately distributed noise to (hyper)edge weights. Given that our optimization problem is NP-hard, we turn our attention to designing efficient approximation algorithms. For the case of uncertain weighted graphs, we provide a 13\frac{1}{3}-approximation algorithm, and a 15\frac{1}{5}-approximation algorithm with near optimal run time. For the case of uncertain weighted hypergraphs, we provide a Ω(1k)\Omega(\frac{1}{k})-approximation algorithm, where kk is the rank of the hypergraph (i.e., any hyperedge includes at most kk nodes), that runs in almost (modulo log factors) linear time. We complement our theoretical results by testing our approximation algorithms on a wide variety of synthetic experiments, where we observe in a controlled setting interesting findings on the trade-off between reward, and risk. We also provide an application of our formulation for providing recommendations of teams that are likely to collaborate, and have high impact.Comment: 25 page

    Algorithms for Vertex-Weighted Matching in Graphs

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    A matching M in a graph is a subset of edges such that no two edges in M are incident on the same vertex. Matching is a fundamental combinatorial problem that has applications in many contexts: high-performance computing, bioinformatics, network switch design, web technologies, etc. Examples in the first context include sparse linear systems of equations, where matchings are used to place large matrix elements on or close to the diagonal, to compute the block triangular decomposition of sparse matrices, to construct sparse bases for the null space or column space of under-determined matrices, and to coarsen graphs in multi-level graph partitioning algorithms. In the first part of this thesis, we develop exact and approximation algorithms for vertex weighted matchings, an under-studied variant of the weighted matching problem. We propose three exact algorithms, three half approximation algorithms, and a two-third approximation algorithm. We exploit inherent properties of this problem such as lexicographical orders, decomposition into sub-problems, and the reachability property, not only to design efficient algorithms, but also to provide simple proofs of correctness of the proposed algorithms. In the second part of this thesis, we describe work on a new parallel half-approximation algorithm for weighted matching. Algorithms for computing optimal matchings are not amenable to parallelism, and hence we consider approximation algorithms here. We extend the existing work on a parallel half approximation algorithm for weighted matching and provide an analysis of its time complexity. We support the theoretical observations with experimental results obtained with MatchBoxP, toolkit designed and implemented in C++ and MPI using modern software engineering techniques. The work in this thesis has resulted in better understanding of matching theory, a functional public-domain software toolkit, and modeling of the sparsest basis problem as a vertex-weighted matching problem

    Bi-Criteria and Approximation Algorithms for Restricted Matchings

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    In this work we study approximation algorithms for the \textit{Bounded Color Matching} problem (a.k.a. Restricted Matching problem) which is defined as follows: given a graph in which each edge ee has a color cec_e and a profit peQ+p_e \in \mathbb{Q}^+, we want to compute a maximum (cardinality or profit) matching in which no more than wjZ+w_j \in \mathbb{Z}^+ edges of color cjc_j are present. This kind of problems, beside the theoretical interest on its own right, emerges in multi-fiber optical networking systems, where we interpret each unique wavelength that can travel through the fiber as a color class and we would like to establish communication between pairs of systems. We study approximation and bi-criteria algorithms for this problem which are based on linear programming techniques and, in particular, on polyhedral characterizations of the natural linear formulation of the problem. In our setting, we allow violations of the bounds wjw_j and we model our problem as a bi-criteria problem: we have two objectives to optimize namely (a) to maximize the profit (maximum matching) while (b) minimizing the violation of the color bounds. We prove how we can "beat" the integrality gap of the natural linear programming formulation of the problem by allowing only a slight violation of the color bounds. In particular, our main result is \textit{constant} approximation bounds for both criteria of the corresponding bi-criteria optimization problem

    Maximum Matching in Turnstile Streams

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    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 22-approximation streaming algorithm can be easily obtained with space O(nlogn)O(n \log n), where nn 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 O(n3/2δ)O(n^{3/2-\delta}) is granted, for every δ>0\delta > 0. Specifically, for every 0ϵ10 \le \epsilon \le 1, we show that in the one-pass turnstile streaming model, in order to compute a O(nϵ)O(n^{\epsilon})-approximation, space Ω(n3/24ϵ)\Omega(n^{3/2 - 4\epsilon}) is required for constant error randomized algorithms, and, up to logarithmic factors, space O(n22ϵ)O( n^{2-2\epsilon} ) is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest

    Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

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    This paper presents an algorithm for estimating the weight of a maximum weighted matching by augmenting any estimation routine for the size of an unweighted matching. The algorithm is implementable in any streaming model including dynamic graph streams. We also give the first constant estimation for the maximum matching size in a dynamic graph stream for planar graphs (or any graph with bounded arboricity) using O~(n4/5)\tilde{O}(n^{4/5}) space which also extends to weighted matching. Using previous results by Kapralov, Khanna, and Sudan (2014) we obtain a polylog(n)\mathrm{polylog}(n) approximation for general graphs using polylog(n)\mathrm{polylog}(n) space in random order streams, respectively. In addition, we give a space lower bound of Ω(n1ε)\Omega(n^{1-\varepsilon}) for any randomized algorithm estimating the size of a maximum matching up to a 1+O(ε)1+O(\varepsilon) factor for adversarial streams
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