[Paper Review] Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor

Bart M. P. Jansen, Marcin Pilipczuk+1 more|arXiv (Cornell University)|Jan 1, 2018

Complexity and Algorithms in Graphs1 citations

TL;DR

This paper presents polynomial Turing kernelization for the k-Path problem in graphs excluding a fixed graph H as a topological minor, using an oracle for bounded-size subproblems. It extends prior results by handling broader graph classes and even graphs with a small vertex modulator to an H-topological-minor-free graph, leveraging structural graph decompositions and oracle-assisted reductions to achieve efficient kernelization in polynomial time with poly(k) oracle queries.

ABSTRACT

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.

Motivation & Objective

• To resolve the open problem of whether k-Path admits a polynomial Turing kernel in broader graph classes beyond bounded-degree and K3,t-minor-free graphs.
• To extend Turing kernelization to graphs that are not H-topological-minor-free themselves, but contain a small vertex modulator whose deletion yields such a graph.
• To develop a win/win strategy based on graph minors and topological minor decomposition to identify reducible separations for kernelization.
• To establish that k-Path admits a polynomial Turing kernel under parameterization by k and |M|, where M is a modulator to H-topological-minor-free graphs.

Proposed method

• Leveraging Robertson and Seymour’s graph minors decomposition and Grohe and Marx’s topological minor decomposition to find tree decompositions of bounded adhesion and width poly(k) in H-topological-minor-free graphs.
• Using a win/win argument: either a k-path exists, or a reducible separation (A,B) is found where A has size poly(k) and is guarded by a small set Z.
• Applying a reduction rule that deletes unmarked vertices in A after oracle queries to mark all possible k-path traversals through A, using the Auxiliary Linkage oracle on N[A].
• Ensuring that the number of components in the decomposition is bounded by poly(k, w, s), so that if no reduction applies, the instance size is polynomial in k, w, and s.
• Using the fact that graphs embeddable in a fixed surface are K3,t-minor-free, and applying known lower bounds on path length in such graphs to infer existence of k-paths.
• Adapting the approach to k-Cycle by first testing for a path of length k², and using Dirac’s theorem to guarantee a long cycle if such a path exists.

Experimental results

Research questions

• RQ1 Does k-Path admit a polynomial Turing kernel in graphs excluding a fixed graph H as a topological minor?
• RQ2 Can polynomial Turing kernelization be achieved for k-Path in graphs that are not H-topological-minor-free, but contain a small vertex modulator to such a class?
• RQ3 Can the win/win strategy based on graph minors decomposition be adapted to find reducible separations in H-topological-minor-free graphs?
• RQ4 What structural properties of H-topological-minor-free graphs allow for efficient oracle-assisted kernelization of k-Path?
• RQ5 Can the techniques be extended to other problems such as k-Cycle or Induced k-Path?

Key findings

• k-Path admits a polynomial Turing kernel in H-topological-minor-free graphs, with runtime k^O(H)(n²m) and k^O(H) · n oracle calls.
• The kernelization extends to graphs with a known vertex modulator M of size bounded polynomially in k, where G−M is H-topological-minor-free, under parameterization by k and |M|.
• The existence of a k-path in H-topological-minor-free graphs can be determined via a win/win: either a k-path exists, or a reducible separation (A,B) is found with |A| = poly(k) and a guard set Z of size O(1).
• The reduction rule deletes at least one vertex from A after a number of oracle queries to the Auxiliary Linkage oracle on G[N[A]], preserving the k-path property.
• If no reduction applies, the instance size is bounded by poly(k, w, s), allowing a single oracle call to solve the problem.
• The approach can be adapted to k-Cycle by first testing for a path of length k², leveraging Dirac’s theorem to infer long cycles when such a path exists.