3,500 research outputs found
New Iterative Algorithms for Weighted Matching
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
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 -approximation algorithm, and a
-approximation algorithm with near optimal run time. For the case
of uncertain weighted hypergraphs, we provide a
-approximation algorithm, where is the rank of the
hypergraph (i.e., any hyperedge includes at most 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
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
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 has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color 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 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
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 -approximation
streaming algorithm can be easily obtained with space , where
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 is
granted, for every . Specifically, for every , we show that in the one-pass turnstile streaming model, in order to compute
a -approximation, space is
required for constant error randomized algorithms, and, up to logarithmic
factors, space 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
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 space which also
extends to weighted matching. Using previous results by Kapralov, Khanna, and
Sudan (2014) we obtain a approximation for general graphs
using space in random order streams, respectively. In
addition, we give a space lower bound of for any
randomized algorithm estimating the size of a maximum matching up to a
factor for adversarial streams
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