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

2. Tata Institute of Fundamental Research, Mumbai
Source:

3. Preserving Distance Between Terminals𝐺
Given
𝐺: undirected, unweighted (𝑛 vertices)
𝑇: terminals (𝑘 vertices)
Typically, 𝑘≪𝑛
Objective
𝐻: subgraph of 𝐺 containing the terminals
𝑑 𝐺 𝑢,𝑣 = 𝑑 𝐻 𝑢,𝑣 ∀ 𝑢, 𝑣∈𝑇
Then 𝐻 is a distance-preserving subgraph of 𝐺 𝑢 𝐻
Terminals
Non-terminals 𝑣

4. Preserving Distance Between Terminals𝐺
Given
𝐺: undirected, unweighted (𝑛 vertices)
𝑇: terminals (𝑘 vertices)
Typically, 𝑘≪𝑛
Objective
𝐻: subgraph of 𝐺 containing the terminals
𝑑 𝐺 𝑢,𝑣 = 𝑑 𝐻 𝑢,𝑣 ∀ 𝑢, 𝑣∈𝑇
Then 𝐻 is a distance-preserving subgraph of 𝐺 𝑢 𝐻
Terminals
Non-terminals 𝑣

5. Preserving Distance Between Terminals𝐺
Given
𝐺: undirected, unweighted (𝑛 vertices)
𝑇: terminals (𝑘 vertices)
Typically, 𝑘≪𝑛
Objective
𝐻: subgraph of 𝐺 containing the terminals
𝑑 𝐺 𝑢,𝑣 = 𝑑 𝐻 𝑢,𝑣 ∀ 𝑢, 𝑣∈𝑇
Then 𝐻 is a distance-preserving subgraph of 𝐺 𝑢 𝐻
Terminals
Non-terminals 𝑣

6. Definition
Branching vertex A vertex 𝑣 is called a branching vertex if deg 𝑣 ≥3. 𝑣 𝑣 𝑣
deg 𝑣 =1
Not a branching
vertex
deg 𝑣 =2
Not a branching
vertex
deg 𝑣 =4
Branching vertex

7. Preserving Distance Between Terminals𝐺
𝑑 𝐺 𝑢,𝑣 = 𝑑 𝐻 𝑢,𝑣 ∀ 𝑢, 𝑣∈𝑇
Solution
Trivial: 𝐻=𝐺
Ideal: 𝐻=𝑇
Optimize
Minimize the number of branching vertices in 𝐻 𝐻
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 𝐺 with 𝑡 branching vertices
Distance-preserving minor of 𝐺 with at most 𝑡+𝑘 vertices ⇒

9. Minors vs Subgraphs
Theorem [Krauthgamer, Zondiner 2012] Every graph on 𝑘 terminal vertices admits a distance-preserving minor with at most 𝑂 𝑘 4 vertices.
They explore various classes of graphs (planar graphs, trees). ⇒
Every graph on 𝑘 terminal vertices has a distance-preserving subgraph with at most 𝑂 𝑘 4 branching vertices.

10. Our Results
Theorem [Upper bound] Every interval graph on 𝑘 terminal vertices has a distance-preserving subgraph with at most 𝑂(𝑘 log 𝑘) branching vertices. Question Is this optimal? Answer Yes.

11. Our Results
Theorem [Lower bound] There exists an interval graph on 𝑘 terminal vertices for which every distance-preserving subgraph has at least Ω(𝑘 log 𝑘) 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 𝑘 terminal vertices has a +𝟏 approximating subgraph having at most 𝑂(𝑘) branching vertices. Theorem [Vertices vs edges] There exists an interval graph on 𝑘 terminal vertices such that every optimal distance-preserving subgraph of 𝐺 has 𝑂(𝑘) branching vertices but Ω(𝑘 log 𝑘) 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
Theorem Every interval graph on 𝑘 terminal vertices admits a distance- preserving subgraph having at most 𝑂(𝑘 log 𝑘) 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.
𝑣 2
𝑣 6
𝑣 8
𝑣 1
𝑣 3
𝑣 5
𝑣 4
𝑣 7

16. Interval Graphs 𝑣 2 𝑣 6 𝑣 1 𝑣 8 𝑣 7 𝑣 3 𝑣 5 𝑣 4 𝑣 2 𝑣 6 𝑣 8 𝑣 3 𝑣 5

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).
𝑣 2
𝑣 6
𝑣 8
𝑣 1
𝑣 3
𝑣 5
𝑣 4
𝑣 7

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 ≥ 3).
𝑣 2
𝑣 6
𝑣 8
𝑣 1
𝑣 3
𝑣 5
𝑣 4
𝑣 7

19. Shortest Paths in Interval Graphs
𝑑 Mario,Peach = ?
Idea Greed 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
𝑑 Mario,Peach =6
Proof Induction 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
Lemma All sources can use the path of the source having the leftmost final interval.
Sources
Destination

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

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

38. Proof of the Upper Bound
Define 𝑇 𝑘 = maximum number of branching vertices in an optimal distance-preserving subgraph of an interval graph with 𝑘 terminal vertices. To prove 𝑇 𝑘 ≤𝑂(𝑘 log 𝑘).
Terminals
Non-terminals 𝐺

39. Proof of the Upper Bound
Define 𝑇 𝑘 = maximum number of branching vertices in an optimal distance-preserving subgraph of an interval graph with 𝑘 terminal vertices. To prove 𝑇 𝑘 ≤𝑂(𝑘 log 𝑘).
𝐺 1
𝐺 2

40. Proof of the Upper Bound
To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction.
𝐺 1
𝐺 2

41. Multiple Sources, Multiple Destinations
To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction.
Sources
Destination
𝐺 1
𝐺 2

42. Multiple Sources, Multiple Destinations
To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction.
Sources
Destination
𝐺 1
𝐺 2

43. Multiple Sources, Multiple Destinations
To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction.
Sources
Destination
𝐺 1
𝐺 2

44. Multiple Sources, Multiple Destinations
To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction.
Sources
Destination
𝐺 1
𝐺 2

45. Multiple Sources, Multiple Destinations
To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction.
Sources
Destination
𝐺 1
𝐺 2

46. Multiple Sources, Multiple Destinations
Shortest paths from terminals in 𝐺 1 can be connected to terminals in 𝐺 2 using 𝑂(𝑘) additional branching vertices.
𝐺 1
𝐺 2

47. Completing the Proof 𝑇 𝑘 ≤𝑇 𝑘 2 +𝑇 𝑘 2 +𝑂(𝑘) 𝐺 1 𝐺 2
Both terminal vertices lie in 𝐺 2
Both terminal vertices lie in 𝐺 1
One terminal lies in 𝐺 1 , other in 𝐺 2
𝑇 𝑘 ≤𝑇 𝑘 2 +𝑇 𝑘 2 +𝑂(𝑘)
𝑇 𝑘 2
𝑇 𝑘 2
𝐺 1
𝐺 2

48. Proof of the Lower Bound
Theorem For every 𝑘, there exists an interval graph on 𝑘 terminal vertices for which every distance-preserving subgraph has at least Ω(𝑘 log 𝑘) branching vertices.

49. The Interval Graph Interval starting points: 1,2,…, 𝑘 2 𝑘=8
Interval starting points: 1,2,…, 𝑘 2
Interval length: 𝑘
𝑘=8

50. The Interval Graph Interval starting points: 1,2,…, 𝑘 2 𝑘=8
Interval starting points: 1,2,…, 𝑘 2
Interval length: 𝑘
𝑘=8

51. A New Representation
Represent each interval by a tile at its starting point.
Each tile is a vertex.
𝑘=8

52. A New Representation
Represent each interval by a tile at its starting point.
Each tile is a vertex.
𝑘=8

53. A New Representation
The terminals are strategically placed at 𝑘 of the 𝑘×𝑘 locations. 𝑘 𝑘 𝑘 𝑘

54. A New Representation Terminals Non-terminals
The terminals are strategically placed at 𝑘 of the 𝑘×𝑘 locations. 𝑘 𝑘 𝑘 𝑘

55. Rearranging the Vertices in a Grid𝑘 𝑘 𝑘 𝑘
View as a 𝑘×𝑘 matrix with 0-1 entries.
Non-terminals are 0, terminals are 1.
Permutation matrix.

56. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111

57. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
000
001
𝑟𝑒 𝑣 =000
010
011
100
101
110
111

58. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
000
001
𝑟𝑒 𝑣 =100
010
011
100
101
110
111

59. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
000
001
𝑟𝑒 𝑣 =010
010
011
100
101
110
111

60. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
000
001
𝑟𝑒 𝑣 =110
010
011
100
101
110
111

61. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
Friends Two terminals are called friends if their column indices differ in exactly one bit.
000
001
010 00 1 1
011
100
101
110
111

62. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
Friends Two terminals are called friends if their column indices differ in exactly one bit.
000
001
010 00 1 1
011
100
101
110
111

63. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
Friends Two terminals are called friends if their column indices differ in exactly one bit.
000
001
010 00 1 1
011
100
101
110
111

64. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
Friends Two terminals are called friends if their column indices differ in exactly one bit.
000
001
010 01
011
100
101
110
111

65. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
Friends Two terminals are called friends if their column indices differ in exactly one bit.
000
001
010 01
011
100
101
110
111

66. The Bit-Reversal Permutation Matrix
000
001
010
011
100
101
110
111
Friends Two terminals are called friends if their column indices differ in exactly one bit.
000
001
010 01
011
100
101
Lemma Preserving distance between a pair of friends requires a unique branching vertex.
110
111

67. Completing the Proof
Lemma Preserving distance between a pair of friends requires a unique branching vertex. +
The number of pairs of friends is 𝑘 log 𝑘 2 ⇒
Theorem Preserving distances between all pairs of friends requires Ω 𝑘 log 𝑘 branching vertices.

68. Conclusion Subsequent Work Extended the result to bi-interval graphs.
A polynomial-time algorithm for optimal single-source distance- preserving subgraphs of interval graphs (NP-complete in general).
Open Problems
Is this problem FPT (with parameter 𝑘)?
Generalizations of interval graphs (perfect graphs, chordal graphs).

69. Thank you!