Twin-width and Polynomial Kernels

Bonnet, Édouard; Kim, Eun Jung; Reinald, Amadeus; Thomassé, Stéphan; Watrigant, Rémi

doi:10.1007/s00453-022-00965-5

Cited by 15 publications

(5 citation statements)

References 46 publications

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“…We show tightness of our bounds on the twin-width of Cartesian, tensor and corona products. We also show tightness of the bound for strong and lexicographical products proved by Bonnet et al (2021a) and Bonnet et al (2022d) respectively. We also show that our bound on rooted products is almost tight.…”

Section: Productsupporting

confidence: 80%

“…Additionally, for graphs of bounded twin-width with the contraction sequence as input, there is a linear time algorithm for triangle counting Kratsch et al (2022), and an FPT-algorithm for maximum independent set Bonnet et al (2021b). Polynomial kernels for several problems Bonnet et al (2022d) are also known which do not require the contraction sequence to be given as input. However, deciding if the twin-width of a graph is at most four is NP-complete Bergé et al (2022).…”

Section: Introductionmentioning

confidence: 99%

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Bounds on the Twin-Width of Product Graphs

Pettersson¹,

Sylvester²

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.

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Section: Productsupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Bounds on the Twin-Width of Product Graphs

Pettersson¹,

Sylvester²

2023

Discrete Mathematics &Amp; Theoretical Computer Science

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show abstract

“…It can make the original nonlinear relationship become linearly separable after the data is mapped to kernel space. Common kernel functions include polynomial kernel [38], gaussian kernel [39] and sigmoid kernel [40]. In this paper, we use gaussian kernel function to map the data, so as to learn the nonlinear relationship in the data.…”

Section: Kernel Function and Graph Laplacianmentioning

confidence: 99%

Piecewise Weighting Function for Collaborative Filtering Recommendation

Zhang¹,

Li²

2022

Preprint

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<p>It is not always reasonable to set a fixed weight value to an attribute (or a variable) because this cannot accurately maintain the similarity between users and leads to decrease the performance of collaborative filtering recommendation algorithms. In this paper, a piecewise weighting method is proposed with the hyper-class representation for improving the collaborative filtering recommendation. Specifically, a kernel function is first applied to map the original data into the kernel space, and to learn the weight of each attribute. And then, the hyper-class representation of the data is constructed to learn the weight of the segmented attribute value (hyper-class) for each attribute, so as to construct a piecewise weighting function. In addition, the piecewise weighting function is applied to calculate the similarity between users for a collaborative filtering recommendation. Finally, experiments are conducted to examine the collaborative filtering recommendation algorithm, and show that the proposed algorithm with piecewise weighting functions is more efficient than the compared algorithm with fixed weight values, in terms of Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and Precision.</p>

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“…This property seems to characterize precisely the graph classes that are regarded as structurally sparse. Since twin width of graphs and binary relational structures has been introduced in [10], it received a lot of attention in algorithmic structural graph theory [4,5,6,7,8,9,20,21,22,29,44]. (We defer the somewhat unwieldy definition of twin width to Section 2.3.)…”

Section: Introductionmentioning

confidence: 99%

Isomorphism Testing for Graphs Excluding Small Minors

2023

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We prove that isomorphism of tournaments of twin width at most k can be decided in time k O(log k) n O(1) . This implies that the isomorphism problem for classes of tournaments of bounded or moderately growing twin width is in polynomial time. By comparison, there are classes of undirected graphs of bounded twin width that are isomorphism complete, that is, the isomorphism problem for the classes is as hard as the general graph isomorphism problem. Twin width is a graph parameter that has been introduced only recently (Bonnet et al., FOCS 2020), but has received a lot of attention in structural graph theory since then. On directed graphs, it is functionally smaller than clique width. We prove that on tournaments (but not on general directed graphs) it is also functionally smaller than directed tree width (and thus, the same also holds for cut width and directed path width). Hence, our result implies that tournament isomorphism testing is also fixed-parameter tractable when parameterized by any of these parameters.Our isomorphism algorithm heavily employs group-theoretic techniques. This seems to be necessary: as a second main result, we show that the combinatorial Weisfeiler-Leman algorithm does not decide isomorphism of tournaments of twin width at most 35 if its dimension is o( √ n). (Throughout this abstract, n is the order of the input graphs.

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Twin-width and Polynomial Kernels

Cited by 15 publications

References 46 publications

Bounds on the Twin-Width of Product Graphs

Bounds on the Twin-Width of Product Graphs

Piecewise Weighting Function for Collaborative Filtering Recommendation

Isomorphism Testing for Graphs Excluding Small Minors

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