We consider a natural variant of the well-known Feedback Vertex Set problem, namely the problem of deleting a small subset of vertices or edges to a full binary tree. This version of the problem is motivated by real-world scenarios that are best modeled by full binary trees. We establish that both versions of the problem are NP-hard, which stands in contrast to the fact that deleting edges to obtain a forest or a tree is equivalent to the problem of finding a minimum cost spanning tree, which can be solved in polynomial time. We also establish that both problems are FPT by the standard parameter.

1. Deleting to Structured Trees
Pratyush Dayal and Neeldhara Misra
IIT Gandhinagar
COCOON 2019
Xian, China — 29th July

2. Overview
Introduction & Summary of Results
Hardness of deleting to ternary trees: a warm-up reduction
Deletion to binary trees - the vertex version
Deletion to binary trees - the edge version
Algorithmic questions & future directions

3. Overview
Introduction & Summary of Results
Hardness of deleting to ternary trees: a warm-up reduction
Deletion to binary trees - the vertex version
Deletion to binary trees - the edge version
Algorithmic questions & future directions

4. Once upon a time*, there was the Feedback Vertex Set problem.
Feedback Vertex Set
Forest
*Classical NP-complete problem

5. Related question (1): delete vertices to obtain a tree.
Tree Deletion Set
Tree
Venkatesh Raman, Saket Saurabh, and Ondrej Suchý. An FPT algorithm for tree deletion set. J. Graph Algorithms Appl., 17(6):615–628, 2013.
Archontia C. Giannopoulou, Daniel Lokshtanov, Saket Saurabh, and Ondrej Suchý. Tree deletion set has a polynomial kernel but no opto(1) approximation.
SIAM J. Discrete Math., 30(3):1371– 1384, 2016.

6. Related question (2): delete vertices to obtain a forest of pathwidth one.
PW-One Deletion Set
Caterpillars
Geevarghese Philip, Venkatesh Raman, and Yngve Villanger. A quartic kernel for pathwidth-one vertex deletion. (WG 2010)
MarekCygan,MarcinPilipczuk,MichalPilipczuk,andJakubOnufryWojtaszczyk.
An improved FPT algorithm and a quadratic kernel for pathwidth one vertex deletion. Algorithmica, 64(1):170–188, 2012.

7. Related question (3): delete vertices to obtain stars of bounded degree.
Star Deletion Set
Stars
Robert Ganian, Fabian Klute, and Sebastian Ordyniak.
On structural parameterizations of the bounded-degree vertex deletion problem (STACS 2018)

8. Our problem: delete vertices to obtain a full binary tree.
FBT Deletion Set
Full
binary tree

9. Rooted full binary tree: every vertex has either two children or no children.
Public domain image (source: Wikipedia)
Phylogenetic trees, polymerization processes, etc.

10. The problem of finding a maximum connected
subgraph satisfying a property π is NP-hard for any
non-trivial and interesting property that is
hereditary on induced subgraphs.
Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
J. ACM, 26(4):618–630, 1979.

11. The problem of finding a maximum connected
subgraph satisfying a property π is NP-hard for any
non-trivial and interesting property that is
hereditary on induced subgraphs.
Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
J. ACM, 26(4):618–630, 1979.
A property is a class of graphs.
We will say that the property is satisfied by, or is true for, a
graph G if G belongs to the class given by the property.

12. The problem of finding a maximum connected
subgraph satisfying a property π is NP-hard for any
non-trivial and interesting property that is
hereditary on induced subgraphs.
A property is said to be non-trivial if
it is true for at least one graph and false for at least one graph.
Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
J. ACM, 26(4):618–630, 1979.

13. The problem of finding a maximum connected
subgraph satisfying a property π is NP-hard for any
non-trivial and interesting property that is
hereditary on induced subgraphs.
A property is said to be interesting if it is true
for arbitrarily large graphs.
Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
J. ACM, 26(4):618–630, 1979.

14. The problem of finding a maximum connected
subgraph satisfying a property π is NP-hard for any
non-trivial and interesting property that is
hereditary on induced subgraphs.
A property is said to be hereditary on induced subgraphs
if the deletion of any node from a graph that has the property
always results in a graph that also has the property.
Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
J. ACM, 26(4):618–630, 1979.

15. The problem of finding a maximum connected
subgraph satisfying a property π is NP-hard for any
non-trivial and interesting property that is
hereditary on induced subgraphs.
Tree Deletion Set
Plug in the property of acyclicity above.
Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
J. ACM, 26(4):618–630, 1979.

16. The problem of finding a maximum connected
subgraph satisfying a property π is NP-hard for any
non-trivial and interesting property that is
hereditary on induced subgraphs.
FBT Deletion Set
The property of being a binary tree is not
hereditary on induced subgraphs.
Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
J. ACM, 26(4):618–630, 1979.

17. The problem of finding a maximum connected
subgraph satisfying a property π is NP-hard for any
non-trivial and interesting property that is
hereditary on induced subgraphs.
FBT Deletion Set
The property of being a binary tree is not
hereditary on induced subgraphs.
Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
J. ACM, 26(4):618–630, 1979.

18. Our Contributions

19. Our Contributions
Deleting optimally to full binary trees by removing
vertices is NP-complete.

20. Our Contributions
Deleting optimally to full binary trees by removing
vertices is NP-complete.
Deleting optimally to full binary trees by removing
edges is NP-complete.

21. Our Contributions
Deleting optimally to full binary trees by removing
vertices is NP-complete.
Deleting optimally to full binary trees by removing
edges is NP-complete.

22. Our Contributions
Deleting optimally to full binary trees by removing
vertices is NP-complete.
Deleting optimally to full binary trees by removing
edges is NP-complete.
Deleting optimally to trees by removing edges is
polynomial time (complement of a MST).

23. Our Contributions
Deleting optimally to full binary trees by removing
vertices is NP-complete.
Deleting optimally to full binary trees by removing
edges is NP-complete.

24. Our Contributions
Deleting optimally to full binary trees by removing
vertices is NP-complete.
Deleting optimally to full binary trees by removing
edges is NP-complete.
Both problems are FPT when parameterized by solution size.

25. Overview
Introduction & Summary of Results
Hardness of deleting to ternary trees: a warm-up reduction
Deletion to binary trees - the vertex version
Deletion to binary trees - the edge version
Algorithmic questions & future directions

26. Deleting to ternary trees
Reduce from Exact Cover by 3-Sets

27. Deleting to ternary trees
Reduce from Exact Cover by 3-Sets

28. Deleting to ternary trees
Reduce from Exact Cover by 3-Sets

29. Deleting to ternary trees
Reduce from Exact Cover by 3-Sets

30. Reduce from Exact Cover by 3-Sets
Deleting to ternary trees

31. Reduce from Exact Cover by 3-Sets
Deleting to ternary trees

32. Reduce from Exact Cover by 3-Sets
Deleting to ternary trees

33. Reduce from Exact Cover by 3-Sets
Deleting to ternary trees

34. Overview
Introduction & Summary of Results
Hardness of deleting to ternary trees: a warm-up reduction
Deletion to binary trees - the vertex version
Deletion to binary trees - the edge version
Algorithmic questions & future directions

35. Vertex deletion to binary trees
Reduce from Multi-Colored Independent Set

36. Vertex deletion to binary trees
Reduce from Multi-Colored Independent Set

37. Vertex deletion to binary trees
Reduce from Multi-Colored Independent Set

38. Vertex deletion to binary trees
Reduce from Multi-Colored Independent Set
Color class
with n vertices
2n leaves in the
constructed instance

39. Vertex deletion to binary trees
Reduce from Multi-Colored Independent Set

40. Vertex deletion to binary trees
Reduce from Multi-Colored Independent Set

41. Vertex deletion to binary trees
Reduce from Multi-Colored Independent Set

42. Vertex deletion to binary trees
Reduce from Multi-Colored Independent Set
Dump a copy of G
on the essential vertices

43. Overview
Introduction & Summary of Results
Hardness of deleting to ternary trees: a warm-up reduction
Deletion to binary trees - the vertex version
Deletion to binary trees - the edge version
Algorithmic questions & future directions

44. Vertex deletion to binary trees
Reduce from Linear Near-Exact (2/2/4)-SAT
Core Clauses
Auxiliary clauses
length four; every shadow variable occurs exactly once with positive polarity
length three; occur in pairs consisting of a main variable and two shadow variables
the shadow variables occur exactly once with negative polarity

45. Vertex deletion to binary trees
Reduce from Linear Near-Exact (2/2/4)-SAT
Core Clauses
Auxiliary clauses
length four; every shadow variable occurs exactly once with positive polarity
length three; occur in pairs consisting of a main variable and two shadow variables
the shadow variables occur exactly once with negative polarity

46. Vertex deletion to binary trees
Reduce from Linear Near-Exact (2/2/4)-SAT
Given a set of clauses as described before,
is there an assignment τ of truth values to the variables such that:
exactly one literal in every core clause
and two literals in every auxiliary clause
evaluate to true under τ?
Details Omitted.

47. Overview
Introduction & Summary of Results
Hardness of deleting to ternary trees: a warm-up reduction
Deletion to binary trees - the vertex version
Deletion to binary trees - the edge version
Algorithmic questions & future directions

48. Algorithmic Questions and Future Directions

49. Algorithmic Questions and Future Directions
These problems are fixed-parameter tractable
with respect to the standard parameter (solution size),
by adapting ideas that have worked for FVS.

50. Algorithmic Questions and Future Directions
These problems are fixed-parameter tractable
with respect to the standard parameter (solution size),
by adapting ideas that have worked for FVS.
Informal Outline

51. Algorithmic Questions and Future Directions
These problems are fixed-parameter tractable
with respect to the standard parameter (solution size),
by adapting ideas that have worked for FVS.
Branch on high-degree vertices and
delete long degree-two paths.
Informal Outline

52. Algorithmic Questions and Future Directions
These problems are fixed-parameter tractable
with respect to the standard parameter (solution size),
by adapting ideas that have worked for FVS.
Branch on high-degree vertices and
delete long degree-two paths.
Branch on short cycles (of length at most log(n)).
Informal Outline

53. Algorithmic Questions and Future Directions

54. Algorithmic Questions and Future Directions
Single-exponential FPT algorithms, perhaps by appropriately adapting
the iterative compression approaches
that have worked well for FVS and related problems.

55. Algorithmic Questions and Future Directions
Single-exponential FPT algorithms, perhaps by appropriately adapting
the iterative compression approaches
that have worked well for FVS and related problems.
Structural Parameters.

56. Algorithmic Questions and Future Directions
Single-exponential FPT algorithms, perhaps by appropriately adapting
the iterative compression approaches
that have worked well for FVS and related problems.
Kernelization.
Structural Parameters.

57. Algorithmic Questions and Future Directions
Single-exponential FPT algorithms, perhaps by appropriately adapting
the iterative compression approaches
that have worked well for FVS and related problems.
Kernelization.
Approximation algorithms.
Structural Parameters.

58. Thank You!
neeldhara.m@iitgn.ac.in