Download presentation
Presentation is loading. Please wait.
Published byOswalda Frei Modified over 6 years ago
1
Turing Kernelization for Finding Long Paths and Cycles in Planar Graphs
Bart M. P. Jansen June 3rd 2016, Algorithms for Optimization Problems Dagstuhl, Germany
2
Finding long paths and cycles
π-Path (π-Cycle) Input: An undirected graph πΊ and an integer π Parameter: π Question: Is there a simple path (cycle) of length at least π? Such a path (cycle) is called a π-path (π-cycle) Generalizes Hamiltonian Path (Cycle), so NP-complete Even on planar graphs of degree at most three π-Path and π-Cycle are fixed-parameter tractable Solvable in π π β
π π(1) time
3
Dan Berret: Find the Longest Path
Woh, oh-oh-oh Find the Longest Path Woh oh-oh If you said P is NP tonight There would still be papers left to write I have a weakness I'm addicted to completeness And I keep searching for the Longest Path The algorithm I would like to see Is of polynomial degree Buts itβs elusive, Nobody has found conclusive Evidence that we can find the Longest Path Dan Berret: Find the Longest Path
4
Parameterized preprocessing
Kernelization models provably effective preprocessing It is a technique to obtain FPT algorithms While π-Path was known to be FPT since 1985, for a long time we did not know whether it has a polynomial kernel In 2008, Bodlaender et al. proved that π-Path and π-Cycle do not admit polynomial kernels unless ππβππππ/ππππ¦ Not even on planar graphs of degree at most three
5
Relaxed notions of preprocessing
For other parameterized problems that do not admit polynomial kernels, researchers found provably effective preprocessing schemes in a slightly different model A reduction to a list of ππππ¦(π)-size instances Are there provably effective preprocessing schemes for π-Path and π-Cycle in such relaxed models? ?
6
Turing kernelization Let πβ Ξ£ β Γβ be a parameterized problem and let π:βββ A Turing kernelization for π of size π is an algorithm that decides any given instance π₯,π of π in time polynomial in π₯ +π when given access to an oracle that for any instance π₯ β² , π β² with π₯ β² , π β² β€π π , decides whether π₯ β² , π β² βπ in a single step
7
Our results Theorem. The π-Path and π-Cycle problems admit polynomial-size Turing kernels when the input graph is planar, or claw-free, or πΎ 3,π‘ -minor-free for some constant π‘, or of constant degree The degree of the polynomial depends on the graph class For Planar π-cycle, Turing kernel of 4 π vertices The difficult part of finding long paths and cycles in these graph classes can be confined to small subtasks
8
Adaptivity The kernel crucially exploits the possibility of an adaptive interaction with the oracle The next queries depend on previous answers Compare to the non-adaptive list of small output instances This is a rare phenomenon, the only other cases known are: π-Independent Set in Bull-free graphs ThomassΓ© et al. [WG 2014] A restricted variant of planar independent set parameterized above lower bounds ZdenΔk DvoΕΓ‘kβs talk [yesterday]
9
Turing kernel for planar π-Cycle
10
Splitting rule for π-Cycle
If there is a connected component of πΊ that is not biconnected, then split it into its biconnected components
11
Long cycles through 2-separators
Claim. Let π΄,π΅βπ(πΊ) such that π΄βͺπ΅=π(πΊ), π΄β©π΅={π’,π£}, and there are no edges between π΄βπ΅ and π΅βπ΄ Let πβπ(πΊ) be the vertices on a longest π’π£ path in πΊ[π΄] If πΊ has a cycle of length β₯π, then: The graph πΊ[π΄] has a cycle of length β₯π, or The graph πΊ[πβͺπ΅] has a cycle of length β₯π
12
Turing reduction rule for π-Longest Cycle
This info can be obtained from the decision oracle for π-Cycle by self-reduction on the ππππ¦(π)-size subgraph πΊ[π΄] If there is a 2-separation π΄,π΅βπ(πΊ) such that π΄β©π΅= π’,π£ is a minimal separator, and π΄ β€ππππ¦(π): If πΊ[π΄] has a cycle of length at least π, output yes If πΊ[π΄] does not have a cycle of length at least π: Query oracle for the vertices π of a longest π’π£ path in πΊ π΄ If π β₯π, then conclude that the answer is yes Else, remove the vertices of π΄βπ from the graph Query the oracle for the instance (πΊ π΄ ,π) with ππππ¦ π vertices
13
Decompose-Query-Reduce
Rule reduces the graph after querying the oracle If every connected component has size ππππ¦(π), we are done Query the oracle for each component, terminate Otherwise, we decompose the input graph into small pieces that interact through vertex sets of size at most two Allows us to find a 2-separation that can be reduced Decomposition step relies on lower bounds on circumference Length of longest simple cycle
14
Circumference of triconnected graphs
Let πΊ be a triconnected graph on π vertices and let β be its circumference If πΊ is planar, then: [Chen & Yu 2002] ββ₯ π log β π 0.63 If πΊ is πΎ 3,π‘ -minor-free, then: [Chen et al. 2012] ββ₯ π‘ π‘β1 β
π log If πΊ is claw-free, then: [Bilinski et al. 2011] ββ₯ π If πΊ has maximum degree at most Ξβ₯4, then: [Chen et al. 2006] ββ₯ π log π πβmaxβ‘(64, 4Ξ+1) A triconnected planar graph with π β π vertices has circumference at least π =π. Triconnected graphs from the considered classes have a cycle (and therefore path) of length π π for some π>0
15
Decomposition into triconnected components
Every graph can be decomposed into triconnected components [Tutte 1966]
16
Decomposition into triconnected components
Every triconnected component is a triconnected topological minor of πΊ Arranged in a tree structure Intersections of adjacent components are minimal separators of πΊ of size at most 2 and define a 2-separation Observation. If a planar graph πΊ has a triconnected component with β₯ π β π vertices, then πΊ has a cycle of length β₯π
17
Modern statement of the Tutte decomposition
Restatement in modern terms (only useful if one does not care for uniqueness of the triconnected components) Every graph πΊ has a tree decomposition such that: Adhesion is β€2 (adjacent bags share β€2 vertices) Every adhesion is a minimal separator Torso of each bag is a triconnected topological minor of πΊ Good to know: Also exists for non-planar graphs Decomposition can be found in linear time Hopcroft & Tarjan, 1973 Theorem. Every graph πΊ has a tree decomposition such that: Adhesion is β€ 2 (adjacent bags share β€ 2 vertices) Every adhesion is a minimal separator Torso of each bag is a triconnected topological minor of πΊ
18
Turing kernelization using the decomposition
Kernel for Planar π-Cycle of size π π βπβ
π π 1.59 If there is a connected component that is not biconnected: Split it into biconnected components, restart If there is a connected component πΊβ with >π(π) vertices: Decompose πΊβ² into triconnected components If some triconnected component πΊββ has β₯ π vertices: πΊ has a cycle of length β₯π: output yes If all triconnected components have < π vertices: We find a 2-separation to apply the reduction rule on
19
Finding a 2-separation to reduce (I)
Recall π π βπβ
π π 1.59 Select lowest node π₯ of decomposition tree whose subtree represents >π(π) vertices of πΊ If deg π₯ β€ π : At most π vertices in component π₯, so child subtrees contain > πβ
π vertices Some child subtree represents more than π and less than π(π) vertices of πΊ Apply the reduction rule to the 2-separation between that child and its parent π₯
20
Finding a 2-separation to reduce (II)
If deg π₯ > π : Triconnected component of π₯ contains at most π vertices Two children π 1 , π 2 of π₯ attach to the same minimal separator {π’,π£} Let π΄ be the vertices represented in π π 1 βͺ π π 2 Let π΅ contain the remaining vertices (and {π’,π£}) Apply the reduction rule to (π΄,π΅)
21
Summary of the kernelization
After decomposing the input graph, if there is a connected component with more than πβ
π π vertices we either find a π-cycle or reduce to a smaller graph Each step decreases the number of vertices or increases the number of biconnected components After a polynomial number of rounds, all connected components have at most π(π) vertices We query the oracle for each of them and decide More careful analysis gives size bound of 4 π vertices π-Cycle has a polynomial Turing kernel on planar graphs
22
Extension to π-Path For π-Path, the reduction rule needs to be updated
There are 6 structurally different ways in which a longest path can cross a 2-separation Reduction rule preserves a maximum-length copy of each
23
Graph theory supporting the kernelization
Proven analogously to existence of good separation This claim shows: If the reduction rule cannot make progress, answer is YES Are there other graph structures for which such a claim holds? Claim. If πΊ is a planar graph on Ξ©( π 2.59 ) vertices not having any separation (π΄,π΅) of order β€2 with: π< π΄ <π π 2.59 then πΊ has a cycle of length at least π.
24
Conclusion The π-Path and π-Cycle problems have polynomial Turing kernels in several restricted graph families In Turing kernelization, reduce using the solutions to small instances of NP-hard subproblems, supplied by the oracle Open problems: What about π-Path in general graphs? Or even chordal graphs? Is there a non-adaptive Turing kernel? Does Exact π-Cycle in planar graphs have a polynomial Turing kernel? THANK YOU!
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.