Deleting to Structured Trees
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The document discusses the complexity of vertex and edge deletion problems related to binary and ternary trees, highlighting NP-completeness, fixed-parameter tractability, and related algorithmic questions. It cites multiple previous works on tree deletion sets and compares various deletion strategies to achieve specific tree structures. Future research directions include exploring single-exponential FPT algorithms and kernelization techniques for these problems.
Deleting to Structured Trees
- 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.
- 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.