Maintaining Approximate Maximum Weighted Matching in Fully Dynamic Graphs

Anand, Abhash; Baswana, Surender; Gupta, Mona; Sen, Sandeep

doi:10.4230/lipics.fsttcs.2012.257

Cited by 8 publications

(5 citation statements)

References 9 publications

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“…We now evaluate the performance of our implementation DynMWMLevel of the meta-algorithm by Crouch, Stubbs, and Williams. First, we use α = (1+ ), so the algorithms under investigation have approximation guarantee of 2(1 + ) 2 . Note that decreasing increases both, the number of levels in the algorithm necessary as well as the work performed by the unweighted matching algorithms on each level of the hierarchy.…”

Section: Methodsmentioning

confidence: 99%

“…The study concludes that in practice an extended random walk-based algorithms should be the method of choice. For the weighted dynamic matching problem, Anand et al [2] propose an algorithm that can maintain an 4.911-approximate dynamic maximum weight matching that runs in amortized O(log n log D) time, where D is the ratio between the highest and the lowest edge weight. Gupta and Peng [27] maintain a (1 + )-approximation under edge insertions/deletions that runs in time O( √ m −2−O(1/ ) log N ) time per update.…”

Section: Preliminariesmentioning

confidence: 99%

“…The main idea of our first dynamic algorithm for the weighted dynamic matching problem is to find random augmenting paths in the graph. To this end, our algorithm uses a random walk process in the graph and then uses dynamic programming on the computed random path P to compute the best possible matching on P. 2 We continue by explaining these concepts in our context and how we handle edge insertions and deletions.…”

Section: Algorithm Dynmwmrandommentioning

confidence: 99%

“…Further, when dealing with large graphs, computing an optimal matching can be too time-consuming and thus one often resorts to approximation to trade running time with solution quality. In recent years, several fully-dynamic algorithms for both exact and approximate MCM [3, 5, 6, 8-11, 14, 25, 27, 32, 33, 42, 43, 46, 48] and MWM [2,26,49] have been proposed. These algorithms exploit previously computed information about the matching to handle graph updates more efficiently than a static recomputation.…”

Section: Introductionmentioning

confidence: 99%

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Fully-dynamic Weighted Matching Approximation in Practice

Angriman¹,

Meyerhenke²,

Schulz³

et al. 2021

Preprint

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Finding large or heavy matchings in graphs is a ubiquitous combinatorial optimization problem. In this paper, we engineer the first non-trivial implementations for approximating the dynamic weighted matching problem. Our first algorithm is based on random walks/paths combined with dynamic programming. The second algorithm has been introduced by Stubbs and Williams without an implementation. Roughly speaking, their algorithm uses dynamic unweighted matching algorithms as a subroutine (within a multilevel approach); this allows us to use previous work on dynamic unweighted matching algorithms as a black box in order to obtain a fully-dynamic weighted matching algorithm. We empirically study the algorithms on an extensive set of dynamic instances and compare them with optimal weighted matchings. Our experiments show that the random walk algorithm typically fares much better than Stubbs/Williams (regarding the time/quality tradeoff), and its results are often not far from the optimum.

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Section: Methodsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Algorithm Dynmwmrandommentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fully-dynamic Weighted Matching Approximation in Practice

Angriman¹,

Meyerhenke²,

Schulz³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…For the weighted dynamic matching problem, Anand et al [11] proposed an algorithm that can maintain an 4.911-approximate dynamic maximum weight matching that runs in amortized O(log log ) time where is the ratio of the weight of the highest weight edge to the weight of the smallest weight edge. Furthermore, a sequence [5,[42][43][44][45] of work on fully dynamic set cover resulted in (1 + )-approximate weighted dynamic matching algorithms for hypergraphs, where is the maximum number of nodes in an hyperedge with O( 2 / 3 + ( / 2 ) log ) amortized and O(( / 3 ) log 2 ( )) worst-case time per operation based on various hierarchical hypergraph decompositions.…”

Section: (Weighted) Matchingmentioning

confidence: 99%

Recent Advances in Fully Dynamic Graph Algorithms

Hanauer¹,

Henzinger²,

Schulz³

2021

Preprint

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In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. Here, we survey recent engineering and theory results in the area of fully dynamic graph algorithms.

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A Batch-dynamic Suitor Algorithm for Approximating Maximum Weighted Matching

Angriman

Boroń

Meyerhenke

2022

ACM J. Exp. Algorithmics

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Matching is a popular combinatorial optimization problem with numerous applications in both commercial and scientific fields. Computing optimal matchings w.r.t. cardinality or weight can be done in polynomial time; still, this task can become infeasible for very large networks. Thus, several approximation algorithms that trade solution quality for a faster running time have been proposed. For networks that change over time, fully dynamic algorithms that efficiently maintain an approximation of the optimal matching after a graph update have been introduced as well. However, no semi- or fully dynamic algorithm for (approximate) maximum weighted matching has been implemented. In this article, we focus on the problem of maintaining a \( 1/2 \) -approximation of a maximum weighted matching (MWM) in fully dynamic graphs. Limitations of existing algorithms for this problem are (i) high constant factors in their time complexity, (ii) the fact that none of them supports batch updates, and (iii) the lack of a practical implementation, meaning that their actual performance on real-world graphs has not been investigated. We propose and implement a new batch-dynamic \( 1/2 \) -approximation algorithm for MWM based on the Suitor algorithm and its local edge domination strategy [Manne and Halappanavar, IPDPS 2014]. We provide a detailed analysis of our algorithm and prove its approximation guarantee. Despite having a worst-case running time of \( \mathcal {O}(n + m) \) for a single graph update, our extensive experimental evaluation shows that our algorithm is much faster in practice. For example, compared to a static recomputation with sequential Suitor , single-edge updates are handled up to \( 10^5\times \) to \( 10^6\times \) faster, while batches of \( 10^4 \) edge updates are handled up to \( 10^2\times \) to \( 10^3\times \) faster.

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Maintaining Approximate Maximum Weighted Matching in Fully Dynamic Graphs

Cited by 8 publications

References 9 publications

Fully-dynamic Weighted Matching Approximation in Practice

Fully-dynamic Weighted Matching Approximation in Practice

Recent Advances in Fully Dynamic Graph Algorithms

A Batch-dynamic Suitor Algorithm for Approximating Maximum Weighted Matching

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