Advanced kernelization algorithms

2015 ◽  
pp. 285-319 ◽  
Author(s):  
Marek Cygan ◽  
Fedor V. Fomin ◽  
Łukasz Kowalik ◽  
Daniel Lokshtanov ◽  
Dániel Marx ◽  
...  
Keyword(s):  
Kernelization Algorithms

FPT and Kernelization Algorithms for the Induced Tree Problem

2021 ◽  
pp. 158-172
Author(s):  
Guilherme Castro Mendes Gomes ◽  
Vinicius F. dos Santos ◽  
Murilo V. G. da Silva ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Kernelization Algorithms

scholarly journals Reconfiguration on Nowhere Dense Graph Classes

10.37236/7458 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Sebastian Siebertz
Keyword(s):  
Minimum Distance ◽  
Dominating Set ◽  
Independent Set ◽  
Closed Graph ◽  
Independent Set Problem ◽  
Graph Classes ◽  
Dominating Set Problem ◽  
Feasible Solutions ◽  
Dense Class ◽  
Kernelization Algorithms

Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to determine whether there exists a sequence $S_1,\ldots,S_n$ of feasible solutions, where $S_1=S_s$, $S_n=S_t$, $|S_i|\leq k\pm 1$, and each $S_{i+1}$ results from $S_i$, $1\leq i<n$, by the addition or removal of a single vertex.We prove that for every nowhere dense class of graphs and for every integer $r\geq 1$ there exists a polynomial $p_r$ such that the reconfiguration variants of the distance-$r$ independent set problem and the distance-$r$ dominating set problem admit kernels of size $p_r(k)$. If $k$ is equal to the size of a minimum distance-$r$ dominating set, then for any fixed $\epsilon>0$ we even obtain a kernel of almost linear size $\mathcal{O}(k^{1+\epsilon})$. We then prove that if a class $\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for some value of $r\geq 1$ the reconfiguration variants of the above problems on $\mathcal{C}$ are $\mathsf{W}[1]$-hard (and in particular we cannot expect the existence of kernelization algorithms). Hence our results show that the limit of tractability for the reconfiguration variants of the distance-$r$ independent set problem and distance-$r$ dominating set problem on subgraph closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.

scholarly journals An overview of kernelization algorithms for graph modification problems

2014 ◽  
Vol 19 (4) ◽  
pp. 346-357 ◽  
Author(s):  
Yunlong Liu ◽  
Jianxin Wang ◽  
Jiong Guo
Keyword(s):  
Graph Modification ◽  
Graph Modification Problems ◽  
Kernelization Algorithms
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