Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k - Path admits a polynomial Tur...

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Vydáno v:Algorithmica Ročník 81; číslo 10; s. 3936 - 3967
Hlavní autoři: Jansen, Bart M. P., Pilipczuk, Marcin, Wrochna, Marcin
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.10.2019
Springer Nature B.V
Témata:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Deletion
Kernels
Mathematics of Computing
Polynomials
Special Issue: Parameterized and Exact Computation
Theory of Computation
Topology
path
Graph minors decomposition
Turing kernelization
ISSN:0178-4617, 1432-0541
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Shrnutí:The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k - Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length  k , using an oracle that answers queries of size  k O ( 1 ) ? We show this can be done when the input graph avoids a fixed graph  H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and  K 3 , t -minor-free graphs. Moreover, we show that k - Path even admits a polynomial Turing kernel when the input graph is not H -topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H -topological-minor-free graph that does not contain a k -path, has a separation that can safely be reduced after communication with the oracle.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00614-4