1. Augmenting Paths, Witnesses and Improved Approximations for Bounded Degree MSTs K. Chaudhuri, S. Rao, S. Riesenfeld, K. Talwar UC Berkeley

2. BDMST : The Problem ► Given:  Graph: G  Edge costs: c e  Degree bound : D ► Find a minimum cost tree that respects the degree bounds 2 1 2 1 2 2 1 1 1 1 2 2 BDMST, D = 3 MST

3. BDMST : The Problem ► Generalization of Minimum Cost Hamiltonian Path ► For general weighted graphs,  No Polynomial-Factor Approximation unless P=NP ► Our Work:  Relax degree bounds to obtain an approximation in cost

4. Previous Results ► [FR94] Unweighted graphs  Additive 1 approximation to degree ► [KR00] Weighted graphs, uniform degree bounds  deg(v) · b(1+  ) D + log b n  cost(T) · (1 + 1/  ) OPT ► [KR03] Non-uniform degree bounds ► [CRRT] Quasipolynomial Running Time  deg(v) · D + log n / log log n  cost(T) · OPT ► [CRRT]Polynomial Running Time  deg(v) · bD + log b n  cost(T) · OPT

5. LP Formulation ► Primal: min  e c e x e  e 2  (v) x e · D  e 2  (v) x e · D x 2 SP G x 2 SP G ► Dual: max v min T (C(T) +  v v (deg T (v) - D)) MST in cost function c uv + u + v v : Penalties on high-degree vertices

6. An Algorithm ► Solve Dual LP  Optimal Penalties v ► Pick MST in cost function (c uv + u + v ) with:  Low maximum degree  Low actual cost

7. An Algorithm ► Solve Dual LP  Optimal Penalties v ► Pick MST in cost function (c uv + u + v ) with:  Maximum degree : D  Real cost : OPT

8. An Algorithm ► Solve Dual LP  Optimal Penalties v ► Pick MST in cost function (c uv + u + v ) with:  Maximum degree : D + O(log n/loglog n)  Real cost : OPT

9. Picking the Right Tree ► T is MST in cost function c uv + u + v with: 1.Max degree : D + O(log n/loglogn) 2.For every vertex v with v > 0, Min degree :D – O(log n/loglog n) ► Theorem: T has  Max degree:D + O(log n/log log n)  Actual cost:OPT

10. MSTDB Problem ► Given:  Graph: G  Edge costs: c e  Degree upper bound: D H  A set of nodes: L  Degree lower bound on L: D L ► Find a MST with  Max degree : D H  Min degree of L : D L ► Or prove that no such tree exists 1 1 2 1 2 1 1 1 1 1 2 1 D H = 3, D L = 2, L = O

11. Previous Work ► [FR94] Unweighted graphs, degree upper bounds  Additive 1 approximation ► [F93] Degree upper bounds  Finds an MST with max degree bD + log b n  Or proves no MST with max degree D exists

12. Our Guarantees ► Given:  Graph: G  Edge costs: c e  Degree upper bound: D H  A set of nodes: L  Degree lower bound on L: D L ► We can find an MST with:  Max Degree: D H + O(log n/log log n)  Min Degree of L : D L – O(log n/log log n) ► Or prove that no MST with given bounds exist

13. Final BDMST Algorithm ► Solve Dual LP  Optimal Penalties v ► Run our MSTDB algorithm :  Cost function : c uv + u + v  Degree upper bound : D  L: Set of nodes with v > 0  Degree lower bound : D ► Theorem: Failure contradicts optimality of v

14. MSTDB Algorithm Outline ► Start with arbitrary MST T 0 ► Phase i: T i-1 : use Augmenting Paths to  Reduce the degree of a “high degree” node  Or increase the degree of a “low degree” node ► Success:  New tree T i ► Failure:  Witness for either D H = d max (T i-1 ) – O(log n/log log n) or D L = d min (L) + O(log n/log log n)

15. Useful Edges and Witness ► Useful edge:  Occurs in some MST of G 1 1 1 1 2 A D C B e f Witness: Structure to show a lower(upper) bound on the degree of any MST

16. High Degree Witness ► High degree Witness:  Center Set : W  Clusters: C 1,.., C k  No useful intercluster edge In any MST: Max Degree (W) ¸ d (|W| + k – 1) / |W| e W

17. Feasible Swaps ► Feasible swap (e,f,T): 1.Tree edge e 2.Non-tree edge f 3.Unique cycle in T [ f contains e 4.c(e) = c(f) ► A feasible swap (e,f,T):  Produces an equal cost tree  Reduces degree of endpoints of e 1 1 1 1 2 A D C B e f

18. Augmenting Paths ► Algorithm:  Construct an augmenting path of feasible swaps

19. W CiCi Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

20. W CiCi Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

21. W CiCi Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

22. W CiCi Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

23. Algorithm Outline ► Start with:  Center Set : W 0  Clusters : connected through W 0 ► In step i:  Find an augmenting path of feasible swaps to improve some v in W i  Failure: Center Set and clusters form a high degree witness

24. Conclusion ► Improved approximation for:  BDMST (Bounded Degree MST)  MSTDB (MST with Degree Bounds) ► New Techniques:  Improved cost bounding techniques based on Edmond’s matching algorithm  Improved degree improvement techniques based on augmenting paths ► Open Question:  Can degree bounds be relaxed to additive constant? ► [FR94] Gives additive 1 for unweighted graphs

25. Questions? Questions?

28. Reduce-Max-Degree Initialize: ► Let  S d = nodes with degree d or more ► Pick d :  |S d-1 | · (log n/log log n) |S d | ► Center Set :  W 0 = S d [ S d-1 ► Initial clusters:  Components when W 0 is deleted from T

29. W CiCi Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

30. Reduce-Max-Degree ► For each useful intercluster edge f :  Find feasible swap (e,f,T) that improves v 2 W t  Remove v from W t  Form new cluster with: ► e ► f ► v ► children clusters of v

31. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

32. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

33. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

34. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

35. Reduce-Max-Degree Termination Conditions: 1. Degree d vertex v removed from W t :  Can find a sequence of swaps to improve v  Number of degree d vertices decreases by one

36. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

37. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

38. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

39. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

40. Tree edge Useful nontree edge Degree d – 1 node Degree d node Low degree node 1 1 1 1 2 2

41. Reduce-Max-Degree Termination Conditions: 1. Degree d vertex v removed from W t :  Can find a set of swaps which improve v  Number of degree d vertices decreases by one 2. No feasible intercluster edges:  Can find witness to show that max degree of any MST is at least d – O(log n/log log n)

42. Obtaining a Witness ► Center Set : W t  |W t | · |S d | ► Clusters:  From deleting W t : ¸ (d-2)|W t |  Lost from merges: · |S d-1 |  Total: ¸ (d-2)|W t | - |S d-1 | ► Witness Quality:  At least (d-2) - O(log n/log log n)

43. Summary ► Improved approximation for:  BDMST (Bounded Degree MST)  MSTDB (MST with Degree Bounds) ► New Techniques:  Improved cost bounding techniques based on Edmond’s matching algorithm  Improved degree improvement techniques based on augmenting paths ► Open Question:  Can degree bounds be relaxed to additive constant? ► [FR94] Gives additive 1 for unweighted graphs

44. A Better Algorithm ► Suppose given D H, we can find an MST with :  Max Degree : 5D H + 2  Or show there is no MST with max degree D H ► BDMST Guarantees:  deg(v) · 10(1+  )D + O(1)  cost(T) · (1+1/  ) OPT ► MSTDB Algorithm uses Push-Relabel

45. Push Relabel[G85, GT86] ► A node has:  A Label  An Excess ► Push:  Push flow from a higher to a lower label along an edge ► Relabel:  Raise the label of a node with no edges to a lower label ► Feasibility:  Node at label L has edges to nodes at label L-1 or above

46. Push Relabel for Max Flow ► Initially:  Source : 1 excess  Sink : 1 deficit  All other nodes: 0 excess ► Push and Relabel until:  No node has any excess  Or there is a label with no nodes A

47. Push-Relabel for MSTDB ► Works for MSTDB with degree upper bounds only ► Each node has:  A label  An excess/deficit degree ► Excess and Deficits:  deg(v) ¸ d : (deg(v) – d + 1) units excess  deg(v) < d-1: (d – 1 – deg(v)) units deficit

48. Push-Relabel for MSTDB ► Push:  A node can transfer degree to a node at a lower label ► Relabel:  Raise the label of a node which cannot transfer degree to any node at a lower level ► Feasibility:  A node at label L can be improved only by nodes at label L-1 or higher

49. Algorithm Outline ► Sparse Label:  Has less than 4 times as many nodes as the number of nodes in all the labels above it ► Algorithm: Push and relabel until:  Either no nodes have any excess  Or there is a sparse label ► Sparse Label ! Witness  Average degree : d/5 - 2

50. Publications & Manuscripts: 1. [CRRT] Improved Approximation Algorithms for Degree Bounded MSTs using Push-Relabel 2. [CRRT] What would Edmonds do? Augmenting Paths and Witnesses for Degree Bounded MSTs 3. [CCWBPK04] Selfish Caching in Distributed Systems : A Game Theoretic Approach, PODC 2004 4. [CGRT03] Paths, Trees and Minimum Latency Tours, FOCS 2003

51. Future Directions: ► Bounded Degree MSTs:  Improve guarantees to: ► deg(v) · B + O(1) ► cost(T) · OPT ► Embedding simple metrics to l 1 with low distortion

52. Questions?

53. Picking the Right Tree Proof: ► We can show that:  C(T D’ ) +  v D’ v (deg T D’ (v) – D) · OPT D ► If deg T D’ (v) ¸ D when D’ v > 0  C(T D’ ) · OPT D

57. Picking the Right Tree [KR00] [Our Work] ► [F93] Max Degree:  bB + log b n ► Max Degree  B + O(log n/log log n)

58. ► Technical slide with equation about how imposing both upper and lower bounds on degrees gives us optimal cost

59. Picking the Right Tree [KR00] [Our Work] ► Guarantees: For  > 0  Max degree: (1 +  )bB + log b n (1 +  )bB + log b n  Max Cost: (1 + 1/  ) OPT (1 + 1/  ) OPT ► Running time:  Polynomial ► Guarantees:  Max degree: B + O(log n/log log n) B + O(log n/log log n)  Max Cost: OPT OPT ► Running Time:  Quasipolynomial

60. ► Augmenting paths and witnesses : MSTs with Degree Bounds

61. LP Formulation max v min T (C(T) +  v v (deg T (v) - B) MST in cost function c uv + u + v ► OPT Dual: Properties: 1.Max degree : B 2.For all v such that v > 0 deg(v) = B

62. LP Formulation ► Primal: min  e c e x e  e 2  (v) x e · B  e 2  (v) x e · B x 2 SP G x 2 SP G ► Dual: max v min T (C(T) +  v v (deg T (v) - B) MST in cost function c uv + u + v

63. LP Formulation ► Primal: min  e c e x e  e 2  (v) x e · B  e 2  (v) x e · B x 2 SP G x 2 SP G ► Dual: max v min T (C(T) +  v v (deg T (v) - B)

64. MSTDB Algorithm Outline ► Start with arbitrary MST T 0 ► In Phase i  d max = max degree (T i-1 )  d min = min degree (L)  S H = all nodes of degree d max – O(log n/log log n) or more  S L = all nodes in L of degree d min + O(log n/log log n) or less ► Try  Improve a node in S H or S L ► Success:New tree T i ► Failure:Witness for D H = d max – O(log n/log log n) and D L = d min + O(log n/log log n)

65. Low Degree Witness Proof: ► Any MST on W, C 1,.., C k has (|W| + k – 1) edges ► At most (2|W| + k – 2) endpoints can be in W ► Average degree of W · b (2|W| + k – 2)/ |W| c