The Power of Linear-Time Data Reduction for Maximum Matching

Mertzios, George B.; Nichterlein, André; Niedermeier, Rolf

doi:10.1007/s00453-020-00736-0

Cited by 29 publications

(24 citation statements)

References 25 publications

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“…In this article we adopt an experimental approach to answering the following question: do the new reduction rules from Kelk and Linz (2020) produce smaller kernels in practice than, say, when only the subtree and chain reductions are applied? This mirrors several recent articles in the algorithmic graph theory literature where the practical effectiveness of kernelization has also been analyzed (Fellows et al 2018;Ferizovic et al 2020;Henzinger et al 2020;Mertzios et al 2020;Alber et al 2006). The question is relevant, since earlier studies of kernelization in phylogenetics have noted that, despite its theoretical importance, in an empirical setting the chain reduction seems to have very limited effect compared to the subtree reduction (Hickey et al 2008;van Iersel et al 2016).…”

Section: Introductionsupporting

confidence: 69%

Reflections on kernelizing and computing unrooted agreement forests

et al. 2021

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Phylogenetic trees are leaf-labelled trees used to model the evolution of species. Here we explore the practical impact of kernelization (i.e. data reduction) on the NP-hard problem of computing the TBR distance between two unrooted binary phylogenetic trees. This problem is better-known in the literature as the maximum agreement forest problem, where the goal is to partition the two trees into a minimum number of common, non-overlapping subtrees. We have implemented two well-known reduction rules, the subtree and chain reduction, and five more recent, theoretically stronger reduction rules, and compare the reduction achieved with and without the stronger rules. We find that the new rules yield smaller reduced instances and thus have clear practical added value. In many cases they also cause the TBR distance to decrease in a controlled fashion, which can further facilitate solving the problem in practice. Next, we compare the achieved reduction to the known worst-case theoretical bounds of $$15k-9$$ 15 k - 9 and $$11k-9$$ 11 k - 9 respectively, on the number of leaves of the two reduced trees, where k is the TBR distance, observing in both cases a far larger reduction in practice. As a by-product of our experimental framework we obtain a number of new insights into the actual computation of TBR distance. We find, for example, that very strong lower bounds on TBR distance can be obtained efficiently by randomly sampling certain carefully constructed partitions of the leaf labels, and identify instances which seem particularly challenging to solve exactly. The reduction rules have been implemented within our new solver Tubro which combines kernelization with an Integer Linear Programming (ILP) approach. Tubro also incorporates a number of additional features, such as a cluster reduction and a practical upper-bounding heuristic, and it can leverage combinatorial insights emerging from the proofs of correctness of the reduction rules to simplify the ILP.

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Section: Introductionsupporting

confidence: 69%

Reflections on kernelizing and computing unrooted agreement forests

et al. 2021

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“…Data reduction algorithms seem to lend themselves well to realization on GPUs since, often, they consist of a fixed set of data reduction rules, applied to different parts of the data. We thus expect that, although previously more often studied in the context of attacking NP-hard problems [8][9][10], parallel kernelization will have a stronger real-world impact in the context of speeding up polynomial-time algorithms on large data sets, such as linear-time data reduction was applied to speed up matching algorithms [42], or in the context of designing parallel fixed-parameter algorithms [9] for P-complete problems, which do not give in to massive parallelization.…”

Section: Discussionmentioning

confidence: 99%

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

Bevern¹,

Kirilin²,

Skachkov³

et al. 2021

Preprint

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The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasibility of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Date) architectures.

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“…, for some ε > 0. This kind of FPT algorithms for polynomial problems have attracted recent attention [5,16,19,20,21]. We stress that Maximum-Cardinality Matching has been proposed in [21] as the "drosophila" of the study of these FPT algorithms in P. We continue advancing in this research direction.…”

Section: :3mentioning

confidence: 92%

The b-Matching problem in distance-hereditary graphs and beyond

Ducoffe

Popa

2021

Discrete Applied Mathematics

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We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recentlyproposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA'17) and graphs of bounded modular-width (Coudert et al., SODA'18). We present such algorithm for bounded split-width graphs -a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((k log 2 k)•(m+n)•log n)-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG'97) and we answer an open question of (Coudert et al., SODA'18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.

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The Power of Linear-Time Data Reduction for Maximum Matching

Cited by 29 publications

References 25 publications

Reflections on kernelizing and computing unrooted agreement forests

Reflections on kernelizing and computing unrooted agreement forests

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

The b-Matching problem in distance-hereditary graphs and beyond

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