A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection

Transcription

1 A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection Kyle Fox Duke University Philip N. Klein Brown University Shay Mozes IDC Herzliya

2 Minimum Graph Bisection

3 Minimum Graph Bisection Evenly partition vertices

4 Minimum Graph Bisection Minimize # crossing edges Evenly partition vertices

5 NP-hard for general graphs [Garey, Johnson, Stockmeyer 76]

6 NP-hard for general graphs [Garey, Johnson, Stockmeyer 76] O(log n)-approximation [Räcke 08]

7 Planar Graphs Many problems that are hard in general have approximation schemes

8 Planar Graphs Many problems that are hard in general have approximation schemes Cut (bipartition) problems in particular become much easier

9 Planar Graphs Many problems that are hard in general have approximation schemes Cut (bipartition) problems in particular become much easier May even exists a poly-time algorithm for minimum bisection

10 Our Results A bicriteria approximation scheme for minimum bisection in planar graphs

11 Our Results A bicriteria approximation scheme for minimum bisection in planar graphs Let OPT be the cost of the minimum bisection.

12 Our Results A bicriteria approximation scheme for minimum bisection in planar graphs Let OPT be the cost of the minimum bisection. Our algorithm returns a bipartition with at least (1/2 - ε) n vertices on each side

13 Our Results A bicriteria approximation scheme for minimum bisection in planar graphs Let OPT be the cost of the minimum bisection. Our algorithm returns a bipartition with at least (1/2 - ε) n vertices on each side and total cost at most (1 + ε) OPT

14 Our Results A bicriteria approximation scheme for minimum bisection in planar graphs Let OPT be the cost of the minimum bisection. Our algorithm returns a bipartition with at least (1/2 - ε) n vertices on each side and total cost at most (1 + ε) OPT in polynomial time, for any constant ε > 0.

15 Our Results A bicriteria approximation scheme for minimum bisection in planar graphs Works with arbitrary non-negative edge costs and non-negative vertex weights

16 Our Results A bicriteria approximation scheme for minimum bisection in planar graphs Works with arbitrary non-negative edge costs and non-negative vertex weights Also gives a bicriteria approximation scheme for minimum b-balanced cut in planar graphs

17 Our Results A bicriteria approximation scheme for minimum bisection in planar graphs Works with arbitrary non-negative edge costs and non-negative vertex weights Also gives a bicriteria approximation scheme for minimum b-balanced cut in planar graphs Previously known: a 2-approximation algorithm that requires b 1/3 [Garg, Saran, Vazirani 99]

18 Approximations Approximation schemes known for many problems in planar graphs

19 Approximations Approximation schemes known for many problems in planar graphs Ours is the first problem involving balance

20 The Framework Planar graph approximations often follow a single framework [Klein 08]

21 The Framework Planar graph approximations often follow a single framework [Klein 08] To use the framework for bisection, all we had to do was provide a construction for the sparsifier

22 Contracting an edge merges its endpoints

23 Sparsifier

24 Sparsifier Contraction with Õ(f(ε) OPT) edges

25 Sparsifier Contraction with Õ(f(ε) OPT) edges Cost of optimal (near-)bisection goes up by at most 1+ε factor

26 Duality Vertices faces Edges edges Faces vertices Edge contraction edge deletion v v* f g f* g* u u*

27 Cut-Cycle Duality

28 Cut-Cycle Duality

29 Cut-Cycle Duality

30 Cut-Cycle Duality

31 Cut-Cycle Duality

32 Cut-Cycle Duality

33 In dual, optimal bisection is a collection of cycles enclosing half the faces

34 In dual, optimal bisection is a collection of cycles enclosing half the faces Contractions in the input graph are deletions in its dual

35 In dual, optimal bisection is a collection of cycles enclosing half the faces Contractions in the input graph are deletions in its dual So in the dual, the sparsifier is a subgraph

36 Goal: Given a planar graph G (the dual), find a subgraph H (the sparsifier) such that

37 Goal: Given a planar graph G (the dual), find a subgraph H (the sparsifier) such that Cost of H is Õ(f(ε) OPT)

38 Goal: Given a planar graph G (the dual), find a subgraph H (the sparsifier) such that Cost of H is Õ(f(ε) OPT) H contains a cheap collection of cycles enclosing (roughly) half the faces

39 Goal: Given a planar graph G (the dual), find a subgraph H (the sparsifier) such that Cost of H is Õ(f(ε) OPT) H contains a cheap collection of cycles enclosing (roughly) half the faces Cheap = (1 + ε) OPT

40 Goal: Given a planar graph G (the dual), find a subgraph H (the sparsifier) such that Cost of H is Õ(f(ε) OPT) H contains a cheap collection of cycles enclosing (roughly) half the faces Cheap = (1 + ε) OPT Roughly = to within (1 + ε) factor

41 : optimal bisection cycle

42 : optimal bisection cycle

43 : optimal bisection cycle

44 : optimal bisection cycle

45 : optimal bisection cycle

46 : optimal bisection cycle

47 : optimal bisection cycle

48 Boundary-to-boundary Spanner [Klein 06]

49 Boundary-to-boundary Spanner [Klein 06]

50 Boundary-to-boundary Shortest path Spanner

51 Boundary-to-boundary Shortest path Spanner Spanner path

52 : optimal bisection cycle

53 : optimal bisection cycle

54 : optimal bisection cycle : skeleton cycle

55 : optimal bisection cycle : skeleton cycle

56 : optimal bisection cycle : skeleton cycle

57 : optimal bisection cycle : skeleton cycle : sparsifier cycle

58 : optimal bisection cycle : skeleton cycle : sparsifier cycle

59 Bisection Algorithm 1. Find a sparsifier with Õ(f(ε) OPT) edges a. Find many skeleton cycles in dual graph b. Add edges for boundary-to-boundary spanner paths 2. Apply remaining steps of sparsifier framework

60 Bisection Algorithm 1. Find a sparsifier with Õ(f(ε) OPT) edges a. Find many skeleton cycles in dual graph b. Add edges for boundary-to-boundary spanner paths 2. Apply remaining steps of sparsifier framework

61 Bisection Algorithm 1. Find a sparsifier with Õ(f(ε) OPT) edges a. Find many skeleton cycles in dual graph b. Add edges for boundary-to-boundary spanner paths 2. Apply remaining steps of sparsifier framework

62 : optimal bisection cycle : skeleton cycle : sparsifier cycle

63 : optimal bisection cycle : skeleton cycle : sparsifier cycle

64 : optimal bisection cycle : skeleton cycle : sparsifier cycle Perturbation cycle

65 Understanding Cycles Need to understand the cost and weight of cycles

66 Understanding Cycles Need to understand the cost and weight of cycles cost(c) := number of edges along the cycle C

67 Understanding Cycles Need to understand the cost and weight of cycles cost(c) := number of edges along the cycle C weight(c) := number of faces enclosed by C

68 Understanding Cycles Need to understand the cost and weight of cycles cost(c) := number of edges along the cycle C weight(c) := number of faces enclosed by C ratio(c) := cost(c) / weight(c)

69 Ratio is the Key Claim: Removing all cycles of high ratio (at least OPT / (εn)) from the optimal solution causes at most εn faces to switch sides

70 Ratio is the Key Claim: Removing all cycles of high ratio (at least OPT / (εn)) from the optimal solution causes at most εn faces to switch sides Proof: Faces outside these cycles don t change sides

71 Ratio is the Key Claim: Removing all cycles of high ratio (at least OPT / (εn)) from the optimal solution causes at most εn faces to switch sides Proof: Faces outside these cycles don t change sides Optimal bisection cycles have total cost OPT

72 Ratio is the Key Claim: Removing all cycles of high ratio (at least OPT / (εn)) from the optimal solution causes at most εn faces to switch sides Proof: Faces outside these cycles don t change sides Optimal bisection cycles have total cost OPT So total weight inside high-ratio cycles is at most OPT * (εn) / OPT = εn

73 Perturbation cycle Can show high-ratio perturbation cycles have total weight O(εn)

74 Perturbation cycle Can show high-ratio perturbation cycles have total weight O(εn) Main technical idea: Guarantee all perturbation cycles have high ratio

75 Perturbation cycle Can show high-ratio perturbation cycles have total weight O(εn) Main technical idea: Guarantee all perturbation cycles have high ratio by exhaustively adding cycles with low ratio to the skeleton.

76 : optimal bisection cycle : skeleton cycle : spanner cycle

77 : optimal bisection cycle : skeleton cycle : spanner cycle

78 : optimal bisection cycle : skeleton cycle : spanner cycle

79 : optimal bisection cycle : skeleton cycle : spanner cycle

80 : optimal bisection cycle : skeleton cycle : spanner cycle

81 : optimal bisection cycle : skeleton cycle : spanner cycle

82 : optimal bisection cycle : skeleton cycle : spanner cycle

83 : optimal bisection cycle : skeleton cycle : spanner cycle

84 : optimal bisection cycle : skeleton cycle : spanner cycle

85 : optimal bisection cycle : skeleton cycle : spanner cycle

86 : optimal bisection cycle : skeleton cycle : sparsifier cycle Perturbation cycle

87 : optimal bisection cycle : skeleton cycle : sparsifier cycle Not in the skeleton high ratio!

88 Spanner Paths Between Different Boundary

89 Spanner Paths Between Different Boundary

90 Spanner Paths Between Different Boundary?

91 Spanner Paths Between Different Boundary? Use PC-clustering [Bateni, Hajiaghayi, Marx 11]

92 Keeping the Skeleton Cheap Claim: If skeleton tree has depth d, then the total cost of the skeleton is O(d/ε OPT)

93 Keeping the Skeleton Cheap Claim: If skeleton tree has depth d, then the total cost of the skeleton is O(d/ε OPT) Proof: Skeleton cycles have ratio at most OPT / (εn)

94 Keeping the Skeleton Cheap Claim: If skeleton tree has depth d, then the total cost of the skeleton is O(d/ε OPT) Proof: Skeleton cycles have ratio at most OPT / (εn) Total weight inside skeleton cycles is O(dn)

95 Keeping the Skeleton Cheap Claim: If skeleton tree has depth d, then the total cost of the skeleton is O(d/ε OPT) Proof: Skeleton cycles have ratio at most OPT / (εn) Total weight inside skeleton cycles is O(dn) So total cost is at most O(dn) * OPT / (εn)

96 : optimal bisection cycle : skeleton cycle : spanner cycle

97 : optimal bisection cycle : skeleton cycle : spanner cycle

98 : optimal bisection cycle : skeleton cycle : spanner cycle

99 Conclusions and Open Problems We found a bicriteria approximation scheme for minimum bisection in planar graphs

100 Conclusions and Open Problems We found a bicriteria approximation scheme for minimum bisection in planar graphs Runs in time n poly(1/ε)

101 Conclusions and Open Problems We found a bicriteria approximation scheme for minimum bisection in planar graphs Runs in time n poly(1/ε) Is there an efficient PTAS, i.e. one with running time f(ε) n O(1)?

102 Conclusions and Open Problems We found a bicriteria approximation scheme for minimum bisection in planar graphs Runs in time n poly(1/ε) Is there an efficient PTAS, i.e. one with running time f(ε) n O(1)? Is there a PTAS that returns a perfectly balanced solution?

103 Conclusions and Open Problems We found a bicriteria approximation scheme for minimum bisection in planar graphs Runs in time n poly(1/ε) Is there an efficient PTAS, i.e. one with running time f(ε) n O(1)? Is there a PTAS that returns a perfectly balanced solution? Is minimum bisection actually poly-time solvable in planar graphs?

104 Thank you!