Twin-width and Polynomial Kernels

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for k -Dominating Set on graphs of twin-width at most 4 would contradict a standard com...

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Vydáno v:Algorithmica Ročník 84; číslo 11; s. 3300 - 3337
Hlavní autoři: Bonnet, Édouard, Kim, Eun Jung, Reinald, Amadeus, Thomassé, Stéphan, Watrigant, Rémi
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.11.2022
Springer Nature B.V
Springer Verlag
Témata:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graphs
Kernels
Lower bounds
Mathematics of Computing
Optimization
Polynomials
Special Issue on Parameterized and Exact Computation (IPEC 2021)
Theory of Computation
Upper bounds
Twin-width
Lower bounds
Dominating Set
Kernelization
ISSN:0178-4617, 1432-0541
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Shrnutí:We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for k -Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected k -Dominating Set and Total k -Dominating Set (albeit with a worse upper bound on the twin-width). The k -Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP ’21], which extends to k -Independent Dominating Set , k -Path , k -Induced Path , k -Induced Matching , etc. On the positive side, we obtain a simple quadratic vertex kernel for Connected k -Vertex Cover and Capacitated k -Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik–Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate O ( k 1.5 ) vertex kernel for  Connected k -Vertex Cover . Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-022-00965-5