Engineering Kernelization for Maximum Cut

Ferizovic, Damir; Hespe, Demian; Lamm, Sebastian; Mnich, Matthias; Schulz, Christian; Strash, Darren

doi:10.1137/1.9781611976007.3

Cited by 9 publications

(7 citation statements)

References 30 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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“…In this article we adopt an experimental approach to answering the following question: do the new reduction rules from [23] 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 [8,10,17,27]. 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 [31].…”

Section: Introductionmentioning

confidence: 55%

Reflections on kernelizing and computing unrooted agreement forests

Wersch¹,

Kelk²,

Linz³

et al. 2020

Preprint

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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 and 11k−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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“…For MaxCut, there are several articles that discuss reduction techniques for unweighted MaxCut. Ferizovic et al [14] provide the practically most powerful collection of such techniques. Lange et al [32] provide techniques for general (weighted) MaxCut instances.…”

Section: Simplifying the Problem: Reduction Techniquesmentioning

confidence: 99%

Faster exact solution of sparse MaxCut and QUBO problems

2023

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The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the state of the art in the exact solution of both problems—by using mathematical programming techniques. The main focus lies on sparse problem instances, although also dense ones can be solved. We enhance several algorithmic components such as reduction techniques and cutting-plane separation algorithms, and combine them in an exact branch-and-cut solver. Furthermore, we provide a parallel implementation. The new solver is shown to significantly outperform existing state-of-the-art software for sparse maximum-cut and quadratic unconstrained binary optimization instances. Furthermore, we improve the best known bounds for several instances from the 7th DIMACS Challenge and the QPLIB, and solve some of them (for the first time) to optimality.

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Engineering Kernelization for Maximum Cut

Cited by 9 publications

References 30 publications

Reflections on kernelizing and computing unrooted agreement forests

Reflections on kernelizing and computing unrooted agreement forests

Reflections on kernelizing and computing unrooted agreement forests

Faster exact solution of sparse MaxCut and QUBO problems

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