# Fast Algorithms for Connectivity Problems in Networks

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1 Fast Algorithms for Connectivity Problems in Networks Steiner Cuts, Gomory-Hu Trees, and Edge Splitting Strand Life Sciences 29 May 2008/ MCDES

2 The Setting A network G of n nodes and m edges. Many nodes, relatively fewer connections. Unweighted edges. Redundancy requirement (k) much smaller than n. Reduce algorithms with time complexity O(n 2 ) or more to Õ(n poly(k))

3 The Setting A network G of n nodes and m edges. Many nodes, relatively fewer connections. Unweighted edges. Redundancy requirement (k) much smaller than n. Reduce algorithms with time complexity O(n 2 ) or more to Õ(n poly(k))

4 The Setting A network G of n nodes and m edges. Many nodes, relatively fewer connections. Unweighted edges. Redundancy requirement (k) much smaller than n. Reduce algorithms with time complexity O(n 2 ) or more to Õ(n poly(k))

5 The Setting A network G of n nodes and m edges. Many nodes, relatively fewer connections. Unweighted edges. Redundancy requirement (k) much smaller than n. Reduce algorithms with time complexity O(n 2 ) or more to Õ(n poly(k))

6 The Setting A network G of n nodes and m edges. Many nodes, relatively fewer connections. Unweighted edges. Redundancy requirement (k) much smaller than n. Reduce algorithms with time complexity O(n 2 ) or more to Õ(n poly(k))

7 s t Edge Connectivity/Min-Cut The maximum number of edge disjoint paths between two vertices. Equals the min-cut separating the 2 vertices Can be computed in O(mk) time via Ford-Fulkerson max flows, k is the min-cut size

8 s t Edge Connectivity/Min-Cut The maximum number of edge disjoint paths between two vertices. Equals the min-cut separating the 2 vertices Can be computed in O(mk) time via Ford-Fulkerson max flows, k is the min-cut size

9 s t Edge Connectivity/Min-Cut The maximum number of edge disjoint paths between two vertices. Equals the min-cut separating the 2 vertices Can be computed in O(mk) time via Ford-Fulkerson max flows, k is the min-cut size

10 Global Edge Connectivity/Min-Cut The minimum edge connectivity over all vertex pairs. A global min-cut Best Deterministic Algorithm: O(mk) by Gabow where k is the global min-cut size; near linear Monte Carlo algorithm by Karger.

11 Global Edge Connectivity/Min-Cut The minimum edge connectivity over all vertex pairs. A global min-cut Best Deterministic Algorithm: O(mk) by Gabow where k is the global min-cut size; near linear Monte Carlo algorithm by Karger.

12 Global Edge Connectivity/Min-Cut The minimum edge connectivity over all vertex pairs. A global min-cut Best Deterministic Algorithm: O(mk) by Gabow where k is the global min-cut size; near linear Monte Carlo algorithm by Karger.

13 Steiner Edge Connectivity/Min-Cut S is a subset of interesting Steiner vertices. The minimum edge connectivity over all vertex pairs from S. The min-cut separating vertices in S Question: Is an O(mk) algorithm possible, where k is the Steiner min-cut?

14 Steiner Edge Connectivity/Min-Cut S is a subset of interesting Steiner vertices. The minimum edge connectivity over all vertex pairs from S. The min-cut separating vertices in S Question: Is an O(mk) algorithm possible, where k is the Steiner min-cut?

15 Steiner Edge Connectivity/Min-Cut S is a subset of interesting Steiner vertices. The minimum edge connectivity over all vertex pairs from S. The min-cut separating vertices in S Question: Is an O(mk) algorithm possible, where k is the Steiner min-cut?

16 The Gomory-Hu Tree A tree which carries min-cut information for all pairs of vertices. n 1 max-flow/min-cut computations suffice. Time taken is O(n 3 ) or more. Question: Is Õ(nm) time possible?

17 The Gomory-Hu Tree A tree which carries min-cut information for all pairs of vertices. n 1 max-flow/min-cut computations suffice. Time taken is O(n 3 ) or more. Question: Is Õ(nm) time possible?

18 The Gomory-Hu Tree A tree which carries min-cut information for all pairs of vertices. n 1 max-flow/min-cut computations suffice. Time taken is O(n 3 ) or more. Question: Is Õ(nm) time possible?

19 The Gomory-Hu Tree A tree which carries min-cut information for all pairs of vertices. n 1 max-flow/min-cut computations suffice. Time taken is O(n 3 ) or more. Question: Is Õ(nm) time possible?

20 Gomory-Hu Trees & Steiner Min-Cuts Compute Min-Cut (say Global) Create two recursive sub-problems General Recursive Problem; Steiner Min-Cut

21 Gomory-Hu Trees & Steiner Min-Cuts Compute Min-Cut (say Global) Create two recursive sub-problems General Recursive Problem; Steiner Min-Cut

22 Edge Splitting Pair edges incident on a vertex and remove it Ensure that pairing conserves the global min-cut of the remaining vertices Is this possible at all? Lovasz/Mader: Yes, for eulerian directed graphs and for even degree vertices in undirected graphs

23 Edge Splitting Pair edges incident on a vertex and remove it Ensure that pairing conserves the global min-cut of the remaining vertices Is this possible at all? Lovasz/Mader: Yes, for eulerian directed graphs and for even degree vertices in undirected graphs

24 Edge Splitting Pair edges incident on a vertex and remove it Ensure that pairing conserves the global min-cut of the remaining vertices Is this possible at all? Lovasz/Mader: Yes, for eulerian directed graphs and for even degree vertices in undirected graphs

25 Edge Splitting Pair edges incident on a vertex and remove it Ensure that pairing conserves the global min-cut of the remaining vertices Is this possible at all? Lovasz/Mader: Yes, for eulerian directed graphs and for even degree vertices in undirected graphs

26 Edge Splitting Pair edges incident on a vertex and remove it Ensure that pairing conserves the global min-cut of the remaining vertices Is this possible at all? Lovasz/Mader: Yes, for eulerian directed graphs and for even degree vertices in undirected graphs

27 Edge Splitting and Steiner Min-Cuts Split off all non-steiner vertices Compute global min-cut for what remains in Õ(mk) time.

28 Edge Splitting and Steiner Min-Cuts Split off all non-steiner vertices Compute global min-cut for what remains in Õ(mk) time.

29 Edge Splitting Complexity How fast can one vertex be split off? O(n 2 ) time Karger, Benczur and Õ(nk 2 ) time Gabow How fast can many vertices be split off? Nothing better than plain multiplication known. Is Õ(n polyk) possible?

30 Edge Splitting Complexity How fast can one vertex be split off? O(n 2 ) time Karger, Benczur and Õ(nk 2 ) time Gabow How fast can many vertices be split off? Nothing better than plain multiplication known. Is Õ(n polyk) possible?

31 Edge Splitting Complexity How fast can one vertex be split off? O(n 2 ) time Karger, Benczur and Õ(nk 2 ) time Gabow How fast can many vertices be split off? Nothing better than plain multiplication known. Is Õ(n polyk) possible?

32 Certifying s t and global min-cuts How can one certify a min-cut? s t min-cut by finding edge-disjoint paths. Global min-cut by Edmonds Arborescences

33 Certifying s t and global min-cuts How can one certify a min-cut? s t min-cut by finding edge-disjoint paths. Global min-cut by Edmonds Arborescences

34 Certifying s t and global min-cuts How can one certify a min-cut? s t min-cut by finding edge-disjoint paths. Global min-cut by Edmonds Arborescences

35 Edmonds Arborescences Direct an undirected graph by orienting each edge in both directions. Arborescence: A spanning tree with edges directed away from an arbitrary root Build as many edge-disjoint arborescences as possible rooted at some chosen vertex r Õ(n 2 ) time construction by Gabow.

36 Edmonds Arborescences Direct an undirected graph by orienting each edge in both directions. Arborescence: A spanning tree with edges directed away from an arbitrary root Build as many edge-disjoint arborescences as possible rooted at some chosen vertex r Õ(n 2 ) time construction by Gabow.

37 Edmonds Arborescences Direct an undirected graph by orienting each edge in both directions. Arborescence: A spanning tree with edges directed away from an arbitrary root Build as many edge-disjoint arborescences as possible rooted at some chosen vertex r Õ(n 2 ) time construction by Gabow.

38 Global Min-Cut Characterizations from Edmonds Arborescences Any subset of vertices not containing the root must have in-degree at least 1 in each tree. Existence of k edge-disjoint arborescences implies global min-cut k

39 Global Min-Cut Characterizations from Edmonds Arborescences Any subset of vertices not containing the root must have in-degree at least 1 in each tree. Existence of k edge-disjoint arborescences implies global min-cut k

40 Global Min-Cut Characterizations from Edmonds Arborescences A subset S of vertices s.t. r S, S is contiguous in all trees, and all unused edges directed into S have both endpoints in S, is a global min-cut Gabow s construction shows that such a set exists when no more edge-disjoint arborescences can be conctructed.

41 Global Min-Cut Characterizations from Edmonds Arborescences A subset S of vertices s.t. r S, S is contiguous in all trees, and all unused edges directed into S have both endpoints in S, is a global min-cut Gabow s construction shows that such a set exists when no more edge-disjoint arborescences can be conctructed.

42 Edmonds Directionless Trees Swaps need to be constrained to maintain the arborescence property. Relax the property that edges are directed away from the root. Instead insist that in-degree of a vertex over all directionless trees equals k Existence of k edge-disjoint directionless trees implies global min-cut k Õ(mk) time construction by Gabow.

43 Edmonds Directionless Trees Swaps need to be constrained to maintain the arborescence property. Relax the property that edges are directed away from the root. Instead insist that in-degree of a vertex over all directionless trees equals k Existence of k edge-disjoint directionless trees implies global min-cut k Õ(mk) time construction by Gabow.

44 Edmonds Directionless Trees Swaps need to be constrained to maintain the arborescence property. Relax the property that edges are directed away from the root. Instead insist that in-degree of a vertex over all directionless trees equals k Existence of k edge-disjoint directionless trees implies global min-cut k Õ(mk) time construction by Gabow.

45 Edmonds Directionless Trees Swaps need to be constrained to maintain the arborescence property. Relax the property that edges are directed away from the root. Instead insist that in-degree of a vertex over all directionless trees equals k Existence of k edge-disjoint directionless trees implies global min-cut k Õ(mk) time construction by Gabow.

46 Global Min-Cut Characterizations from Edmonds Directionless Trees A subset S of vertices s.t. r S, S is contiguous in all trees, and all unused edges directed into S having both endpoints in S, is a global min-cut Gabow s construction shows that such a set exists when no more edge-disjoint directionless trees can be constructed.

47 Global Min-Cut Characterizations from Edmonds Directionless Trees A subset S of vertices s.t. r S, S is contiguous in all trees, and all unused edges directed into S having both endpoints in S, is a global min-cut Gabow s construction shows that such a set exists when no more edge-disjoint directionless trees can be constructed.

48 Edmonds Directionless Trees Construction Algorithm Suppose i trees have been constructed and the i + 1th tree is a forest. For each component in the i + 1th tree, run a closure algorithm to identify a subset S of vertices not containing the root, contiguous in all trees, and with all unused edges directed into S having both endpoints in S. If no such subset can be found then a sequence of swaps results in components in the i + 1th tree connecting up. Õ(m) time for the i + 1th tree so Õ(mk) overall; and m can be made O(nk) via Nagamochi-Ibaraki

49 Edmonds Directionless Trees Construction Algorithm Suppose i trees have been constructed and the i + 1th tree is a forest. For each component in the i + 1th tree, run a closure algorithm to identify a subset S of vertices not containing the root, contiguous in all trees, and with all unused edges directed into S having both endpoints in S. If no such subset can be found then a sequence of swaps results in components in the i + 1th tree connecting up. Õ(m) time for the i + 1th tree so Õ(mk) overall; and m can be made O(nk) via Nagamochi-Ibaraki

50 Edmonds Directionless Trees Construction Algorithm Suppose i trees have been constructed and the i + 1th tree is a forest. For each component in the i + 1th tree, run a closure algorithm to identify a subset S of vertices not containing the root, contiguous in all trees, and with all unused edges directed into S having both endpoints in S. If no such subset can be found then a sequence of swaps results in components in the i + 1th tree connecting up. Õ(m) time for the i + 1th tree so Õ(mk) overall; and m can be made O(nk) via Nagamochi-Ibaraki

51 Edmonds Directionless Trees Construction Algorithm Suppose i trees have been constructed and the i + 1th tree is a forest. For each component in the i + 1th tree, run a closure algorithm to identify a subset S of vertices not containing the root, contiguous in all trees, and with all unused edges directed into S having both endpoints in S. If no such subset can be found then a sequence of swaps results in components in the i + 1th tree connecting up. Õ(m) time for the i + 1th tree so Õ(mk) overall; and m can be made O(nk) via Nagamochi-Ibaraki

52 Steiner Min-Cut Algorithm Split-off non-steiner (black) vertices via arbitrary pairing of edges (the global min-cut of the remaining white Steiner vertices could drop in the process) Each edge is actually a path with internal blacks Run Gabow s construction of Edmonds directionless trees on this new graph comprising only whites (ignore blacks)

53 Steiner Min-Cut Algorithm Split-off non-steiner (black) vertices via arbitrary pairing of edges (the global min-cut of the remaining white Steiner vertices could drop in the process) Each edge is actually a path with internal blacks Run Gabow s construction of Edmonds directionless trees on this new graph comprising only whites (ignore blacks)

54 Steiner Min-Cut Algorithm Split-off non-steiner (black) vertices via arbitrary pairing of edges (the global min-cut of the remaining white Steiner vertices could drop in the process) Each edge is actually a path with internal blacks Run Gabow s construction of Edmonds directionless trees on this new graph comprising only whites (ignore blacks)

55 Steiner Min-Cut Algorithm Suppose this culminates in finding a set S of vertices not containing the root, contiguous in all trees, and with all unused edges directed into S having both endpoints in S; S need not be a Steiner min-cut because it need not be contiguous when considering blacks

56 Steiner Min-Cut Algorithm Suppose this culminates in finding a set S of vertices not containing the root, contiguous in all trees, and with all unused edges directed into S having both endpoints in S; S need not be a Steiner min-cut because it need not be contiguous when considering blacks

57 Steiner Min-Cut Algorithm Contd We introduce a mating operation (i.e., revision of pairing) whenever contiguity on blacks is violated This allows closure computation to continue And if black contiguity holds, then S is indeed a Steiner min-cut. And we show how mates can be performed efficiently so the total time stays Õ(nk 2 ), where k is the Steiner min-cut Several technicalities.

58 Steiner Min-Cut Algorithm Contd We introduce a mating operation (i.e., revision of pairing) whenever contiguity on blacks is violated This allows closure computation to continue And if black contiguity holds, then S is indeed a Steiner min-cut. And we show how mates can be performed efficiently so the total time stays Õ(nk 2 ), where k is the Steiner min-cut Several technicalities.

59 Steiner Min-Cut Algorithm Contd We introduce a mating operation (i.e., revision of pairing) whenever contiguity on blacks is violated This allows closure computation to continue And if black contiguity holds, then S is indeed a Steiner min-cut. And we show how mates can be performed efficiently so the total time stays Õ(nk 2 ), where k is the Steiner min-cut Several technicalities.

60 Steiner Min-Cut Algorithm Contd We introduce a mating operation (i.e., revision of pairing) whenever contiguity on blacks is violated This allows closure computation to continue And if black contiguity holds, then S is indeed a Steiner min-cut. And we show how mates can be performed efficiently so the total time stays Õ(nk 2 ), where k is the Steiner min-cut Several technicalities.

61 Splitting-off Vertices in Undirected Graphs Goal: Split-off a specified subset of vertices so global min-cut of the remaining vertices is preserved. Algorithm as above, split-off with arbitrary pairings and then revise pairing via mating

62 Splitting-off Vertices in Undirected Graphs Goal: Split-off a specified subset of vertices so global min-cut of the remaining vertices is preserved. Algorithm as above, split-off with arbitrary pairings and then revise pairing via mating

63 Splitting-off Vertices in Undirected Graphs Complication: this is done on the directed version of the undirected graph; for true undirected splitting-off whenever two directed edges are paired, there reverse edges must be paired as well Coupled Matings required and can be done in Õ(nk 2 ) time.

64 Splitting-off Vertices in Undirected Graphs Complication: this is done on the directed version of the undirected graph; for true undirected splitting-off whenever two directed edges are paired, there reverse edges must be paired as well Coupled Matings required and can be done in Õ(nk 2 ) time.

65 Faster Edmonds Arborescence Construction Goal: Construct k arborescences where k is the global min-cut Algo: Split-off a constant fraction of the vertices, recursively build arborescences for the rest, then put back the split-off vertices Putting back requires that the vertices split-off are independent By Turan s theorem, there is a n/k sized independent set So time taken is Õ(nk 3 ).

66 Faster Edmonds Arborescence Construction Goal: Construct k arborescences where k is the global min-cut Algo: Split-off a constant fraction of the vertices, recursively build arborescences for the rest, then put back the split-off vertices Putting back requires that the vertices split-off are independent By Turan s theorem, there is a n/k sized independent set So time taken is Õ(nk 3 ).

67 Faster Edmonds Arborescence Construction Goal: Construct k arborescences where k is the global min-cut Algo: Split-off a constant fraction of the vertices, recursively build arborescences for the rest, then put back the split-off vertices Putting back requires that the vertices split-off are independent By Turan s theorem, there is a n/k sized independent set So time taken is Õ(nk 3 ).

68 Faster Edmonds Arborescence Construction Goal: Construct k arborescences where k is the global min-cut Algo: Split-off a constant fraction of the vertices, recursively build arborescences for the rest, then put back the split-off vertices Putting back requires that the vertices split-off are independent By Turan s theorem, there is a n/k sized independent set So time taken is Õ(nk 3 ).

69 Faster Edmonds Arborescence Construction Goal: Construct k arborescences where k is the global min-cut Algo: Split-off a constant fraction of the vertices, recursively build arborescences for the rest, then put back the split-off vertices Putting back requires that the vertices split-off are independent By Turan s theorem, there is a n/k sized independent set So time taken is Õ(nk 3 ).

70 Gomory-Hu Tree Construction Find global min-cut Create two Steiner min-cut sub-problems Consider computation tree

71 Gomory-Hu Tree Construction Find global min-cut Create two Steiner min-cut sub-problems Consider computation tree

72 Gomory-Hu Tree Construction Find global min-cut Create two Steiner min-cut sub-problems Consider computation tree

73 Gomory-Hu Tree Construction Consider a leftgoing path in the computation tree New small cuts are identified These are shrunk into single black vertices and split-off So split vertices as they are discovered; vertices to be split are not available in advance

74 Gomory-Hu Tree Construction Consider a leftgoing path in the computation tree New small cuts are identified These are shrunk into single black vertices and split-off So split vertices as they are discovered; vertices to be split are not available in advance

75 Gomory-Hu Tree Construction Consider a leftgoing path in the computation tree New small cuts are identified These are shrunk into single black vertices and split-off So split vertices as they are discovered; vertices to be split are not available in advance

76 Gomory-Hu Tree Construction Consider a leftgoing path in the computation tree New small cuts are identified These are shrunk into single black vertices and split-off So split vertices as they are discovered; vertices to be split are not available in advance

77 Gomory-Hu Tree Construction When a new small cut (black vertex) is discovered, edges incident on it must be paired up for splitting-off Doing so may disconnect existing trees Solution: Directionless Splitting So one leftgoing path takes O(mn) time

78 Gomory-Hu Tree Construction When a new small cut (black vertex) is discovered, edges incident on it must be paired up for splitting-off Doing so may disconnect existing trees Solution: Directionless Splitting So one leftgoing path takes O(mn) time

79 Gomory-Hu Tree Construction When a new small cut (black vertex) is discovered, edges incident on it must be paired up for splitting-off Doing so may disconnect existing trees Solution: Directionless Splitting So one leftgoing path takes O(mn) time

80 Gomory-Hu Tree Construction When a new small cut (black vertex) is discovered, edges incident on it must be paired up for splitting-off Doing so may disconnect existing trees Solution: Directionless Splitting So one leftgoing path takes O(mn) time

81 Gomory-Hu Tree Construction What is the time overall? Partition computation tree into layers The total number of edges in a layer is O(m) (a bit tricky) How many layers are there?

82 Gomory-Hu Tree Construction What is the time overall? Partition computation tree into layers The total number of edges in a layer is O(m) (a bit tricky) How many layers are there?

83 Gomory-Hu Tree Construction What is the time overall? Partition computation tree into layers The total number of edges in a layer is O(m) (a bit tricky) How many layers are there?

84 Gomory-Hu Tree Construction What is the time overall? Partition computation tree into layers The total number of edges in a layer is O(m) (a bit tricky) How many layers are there?

85 Gomory-Hu Trees Construction Choose the root for tree construction randomly O(log n) layers The Gomory-Hu tree can be constructed in Õ(nm) time

86 Gomory-Hu Trees Construction Choose the root for tree construction randomly O(log n) layers The Gomory-Hu tree can be constructed in Õ(nm) time

87 Gomory-Hu Trees Construction Choose the root for tree construction randomly O(log n) layers The Gomory-Hu tree can be constructed in Õ(nm) time

88 The Edge Orientation Problem Given a graph with global min-cut 2k Orient the edges so the resulting directed graph has global min-cut k Challenge: Odd degree vertices cannot be split-off time We show that a minimal graph has sufficiently many even degree vertices Õ(npoly(k)) time algorithm.

89 The Edge Orientation Problem Given a graph with global min-cut 2k Orient the edges so the resulting directed graph has global min-cut k Challenge: Odd degree vertices cannot be split-off time We show that a minimal graph has sufficiently many even degree vertices Õ(npoly(k)) time algorithm.

90 The Edge Orientation Problem Given a graph with global min-cut 2k Orient the edges so the resulting directed graph has global min-cut k Challenge: Odd degree vertices cannot be split-off time We show that a minimal graph has sufficiently many even degree vertices Õ(npoly(k)) time algorithm.

91 The Edge Orientation Problem Given a graph with global min-cut 2k Orient the edges so the resulting directed graph has global min-cut k Challenge: Odd degree vertices cannot be split-off time We show that a minimal graph has sufficiently many even degree vertices Õ(npoly(k)) time algorithm.

92 The Edge Orientation Problem Given a graph with global min-cut 2k Orient the edges so the resulting directed graph has global min-cut k Challenge: Odd degree vertices cannot be split-off time We show that a minimal graph has sufficiently many even degree vertices Õ(npoly(k)) time algorithm.

93 The Survivable Network Design Problem Given a weighted graph Each vertex v has a demand d; v must be connected to every other vertex with at least d edge-disjoint paths. Problem: Find the least weight solution The first sub-quadratic time implementation of the Williamson, Goemans, Mihail, Vazirani algorithm

94 The Survivable Network Design Problem Given a weighted graph Each vertex v has a demand d; v must be connected to every other vertex with at least d edge-disjoint paths. Problem: Find the least weight solution The first sub-quadratic time implementation of the Williamson, Goemans, Mihail, Vazirani algorithm

95 The Survivable Network Design Problem Given a weighted graph Each vertex v has a demand d; v must be connected to every other vertex with at least d edge-disjoint paths. Problem: Find the least weight solution The first sub-quadratic time implementation of the Williamson, Goemans, Mihail, Vazirani algorithm

96 The Survivable Network Design Problem Given a weighted graph Each vertex v has a demand d; v must be connected to every other vertex with at least d edge-disjoint paths. Problem: Find the least weight solution The first sub-quadratic time implementation of the Williamson, Goemans, Mihail, Vazirani algorithm

97 Open Problems Combine splitting-off and sampling (a la Karger) to obtain a Õ(nk) time algorithm for exact/approximate Steiner min-cut Las-Vegas Õ(nk) time algorithm for global Min-Cut

98 Open Problems Combine splitting-off and sampling (a la Karger) to obtain a Õ(nk) time algorithm for exact/approximate Steiner min-cut Las-Vegas Õ(nk) time algorithm for global Min-Cut

99 Collaborators and References Series of joint works with Richard Cole, Anand Balghat, Kavitha T, Debmalya Panigrahy STOC 2003, 2007, SODA 2007, SODA 2008

100 Collaborators and References Series of joint works with Richard Cole, Anand Balghat, Kavitha T, Debmalya Panigrahy STOC 2003, 2007, SODA 2007, SODA 2008

101 And Finally... THANK YOU

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