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Distance-preserving Subgraphs of Interval Graphs

Explore distance-preserving subgraphs in interval graphs, focusing on preserving distance between terminals. Learn about branching, graph classes, spanners, and solvability.

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Distance-preserving Subgraphs of Interval Graphs

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  1. Distance-preserving Subgraphsof Interval Graphs Tata Institute of Fundamental Research, Mumbai Kshitij Gajjar* JaikumarRadhakrishnan ESA 2017, Vienna

  2. Tata Institute of Fundamental Research, Mumbai Source: http://www.tifr.res.in

  3. Preserving Distance Between Terminals Given • undirected, unweighted ( vertices) • terminals ( vertices) • Typically, Objective • subgraph of containing the terminals • Then is a distance-preserving subgraph of Terminals Non-terminals

  4. Preserving Distance Between Terminals Given • undirected, unweighted ( vertices) • terminals ( vertices) • Typically, Objective • subgraph of containing the terminals • Then is a distance-preserving subgraph of Terminals Non-terminals

  5. Preserving Distance Between Terminals Given • undirected, unweighted ( vertices) • terminals ( vertices) • Typically, Objective • subgraph of containing the terminals • Then is a distance-preserving subgraph of Terminals Non-terminals

  6. Definition Branching vertexA vertex is called a branching vertex if . Not a branching vertex Not a branching vertex Branching vertex

  7. Preserving Distance Between Terminals Solution • Trivial: • Ideal: Optimize • Minimize the number of branching vertices in Terminals Non-terminals

  8. Related Work • Graph homeomorphisms [Fortune, Hopcroft, Wyllie 1980] • Graph compression [Feder, Motwani 1995] • Graph spanners [Peleg, Schaffer 1989] • Steiner point removal [Gupta 2001] • Vertex sparsification [Leighton, Moitra 2010] • … Distance-preserving subgraph of with branching vertices Distance-preservingminor of with at most vertices

  9. Minors vs Subgraphs Theorem [Krauthgamer, Zondiner 2012] Every graph on terminal vertices admits a distance-preserving minorwith at most vertices. They explore various classes of graphs (planar graphs, trees). Every graph on terminal vertices has a distance-preserving subgraph with at most branching vertices.

  10. Our Results Theorem[Upper bound] Every interval graph on terminal vertices has a distance-preserving subgraph with at most branching vertices. QuestionIs this optimal? AnswerYes.

  11. Our Results Theorem[Lower bound] There exists an interval graph on terminal vertices for which every distance-preserving subgraph has at least branching vertices.

  12. Our Results (continued) Theorem[Solvability] Finding the optimal distance-preserving subgraph of a graph is NP-complete. Theorem[Additive approximation] Every interval graph on terminal vertices has a approximating subgraph having at most branching vertices. Theorem[Vertices vs edges] There exists an interval graph on terminal vertices such that every optimal distance-preserving subgraph of has branching vertices but branching edges.

  13. Plan for this talk • Proof sketch for the upper bound. • Proof sketch for the lower bound.

  14. Proof of the Upper Bound TheoremEvery interval graph on terminal vertices admits a distance-preserving subgraph having at most branching vertices.

  15. Interval Graphs Definition An interval graph is the intersection graph of a family of intervals on the real line. Intervals are ordered from left to right on the basis of their right end point. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

  16. Interval Graphs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

  17. The Shipping Problem • Intervals represent cargo ships docking at a seaport (vertices). • Certain ships need to transfer containers to and from one another (terminal vertices). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

  18. The Shipping Problem • Minimize the number of transfers per container (shortest paths). • Minimize the number of ships dealing with multiple transfers (vertices of degree ). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

  19. Shortest Paths in Interval Graphs IdeaGreed is good

  20. Shortest Paths in Interval Graphs

  21. Shortest Paths in Interval Graphs

  22. Shortest Paths in Interval Graphs

  23. Shortest Paths in Interval Graphs

  24. Shortest Paths in Interval Graphs

  25. Shortest Paths in Interval Graphs

  26. Shortest Paths in Interval Graphs

  27. Shortest Paths in Interval Graphs

  28. Shortest Paths in Interval Graphs

  29. Shortest Paths in Interval Graphs

  30. Shortest Paths in Interval Graphs

  31. Shortest Paths in Interval Graphs

  32. Shortest Paths in Interval Graphs ProofInduction on path length Such a path is called a greedy shortest path.

  33. Multiple Sources, One Destination Consider the final interval on the greedy shortest path of each source before the destination. Sources Destination

  34. Multiple Sources, One Destination Consider the final interval on the greedy shortest path of each source before the destination. Sources Destination

  35. Multiple Sources, One Destination LemmaAll sources can use the path of the source having the leftmost final interval. Sources Destination

  36. Multiple Sources, One Destination LemmaAll sources can use the path of the source having the leftmost final interval. Sources Destination

  37. Multiple Sources, One Destination LemmaAll sources can use the path of the source having the leftmost final interval. CorollaryIf sources are interconnected, destination requires one vertex to connect to all the sources. Sources Destination

  38. Proof of the Upper Bound Define maximum number of branching vertices in an optimaldistance-preserving subgraph of an interval graph with terminal vertices. To prove. Terminals Non-terminals

  39. Proof of the Upper Bound Define maximum number of branching vertices in an optimaldistance-preserving subgraph of an interval graph with terminal vertices. To prove.

  40. Proof of the Upper Bound To prove. Proof By induction.

  41. Multiple Sources, Multiple Destinations To prove. Proof By induction. Sources Destination

  42. Multiple Sources, Multiple Destinations To prove. Proof By induction. Sources Destination

  43. Multiple Sources, Multiple Destinations To prove. Proof By induction. Sources Destination

  44. Multiple Sources, Multiple Destinations To prove. Proof By induction. Sources Destination

  45. Multiple Sources, Multiple Destinations To prove. Proof By induction. Sources Destination

  46. Multiple Sources, Multiple Destinations Shortest paths from terminals in can be connected to terminals in using additional branching vertices.

  47. Completing the Proof Both terminal vertices lie in Both terminal vertices lie in One terminal lies in other in

  48. Proof of the Lower Bound TheoremFor every , there exists an interval graph on terminal vertices for which every distance-preserving subgraph has at least branching vertices.

  49. The Interval Graph 1 2 3 4 5 6 7 8 9 . . . . . . 32 • Interval starting points:

  50. The Interval Graph 1 2 3 4 5 6 7 8 9 . . . . . . 32 • Interval starting points:

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