Augmenting Paths, Witnesses and Improved Approximations for Bounded Degree MSTs

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Augmenting Paths, Witnesses and Improved
Approximations for Bounded Degree MSTs

• K. Chaudhuri, S. Rao, S. Riesenfeld,
• K. Talwar
• UC Berkeley

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BDMST The Problem

• Given
• Graph G
• Edge costs ce
• Degree bound D
• Find a minimum cost tree that respects the degree
  bounds

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MST
BDMST, D 3
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BDMST The Problem

• Generalization of Minimum Cost Hamiltonian Path
• For general weighted graphs,
• No Polynomial-Factor Approximation unless PNP
• Our Work
• Relax degree bounds to obtain an approximation in
  cost

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Previous Results

• FR94 Unweighted graphs
• Additive 1 approximation to degree
• KR00 Weighted graphs, uniform degree bounds
• deg(v) b(1?) D logb 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
• CRRTPolynomial Running Time
• deg(v) bD logb n
• cost(T) OPT

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LP Formulation

• Primal
• min ?e ce xe
• ?e 2 ?(v) xe D
• x 2 SPG
• Dual

max?v
minT (C(T) ?v ?v(degT(v)
- D))
MST in cost function cuv ?u ?v
?v Penalties on high-degree vertices
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An Algorithm

• Solve Dual LP
• Optimal Penalties ?v
• Pick MST in cost function (cuv ?u ?v) with
• Low maximum degree
• Low actual cost

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An Algorithm

• Solve Dual LP
• Optimal Penalties ?v
• Pick MST in cost function (cuv ?u ?v) with
• Maximum degree D
• Real cost OPT

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An Algorithm

• Solve Dual LP
• Optimal Penalties ?v
• Pick MST in cost function (cuv ?u ?v) with
• Maximum degree D O(log n/loglog n)
• Real cost OPT

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Picking the Right Tree

• T is MST in cost function cuv ?u ?v with
• Max degree D O(log n/loglogn)
• For every vertex v with ?v gt 0,
• Min degree D O(log n/loglog n)
• Theorem T has
• Max degree D O(log n/log log n)
• Actual cost OPT

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MSTDB Problem

• Given
• Graph G
• Edge costs ce
• Degree upper bound DH
• A set of nodes L
• Degree lower bound on L DL
• Find a MST with
• Max degree DH
• Min degree of L DL
• Or prove that no such tree exists

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DH 3, DL 2, L O
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Previous Work

• FR94 Unweighted graphs, degree upper bounds
• Additive 1 approximation
• F93 Degree upper bounds
• Finds an MST with max degree bD logb n
• Or proves no MST with max degree D exists

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Our Guarantees

• Given
• Graph G
• Edge costs ce
• Degree upper bound DH
• A set of nodes L
• Degree lower bound on L DL
• We can find an MST with
• Max Degree DH O(log n/log log n)
• Min Degree of L DL O(log n/log log n)
• Or prove that no MST with given bounds exist

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Final BDMST Algorithm

• Solve Dual LP
• Optimal Penalties ?v
• Run our MSTDB algorithm
• Cost function cuv ?u ?v
• Degree upper bound D
• L Set of nodes with ?v gt 0
• Degree lower bound D
• Theorem Failure contradicts optimality of ?v

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MSTDB Algorithm Outline

• Start with arbitrary MST T0
• Phase i Ti-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 Ti
• Failure
• Witness for either DH dmax(Ti-1) O(log n/log
  log n) or DL dmin(L) O(log n/log log n)

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Useful Edges and Witness
Witness Structure to show a lower(upper) bound
on the degree of any MST

• Useful edge
• Occurs in some MST of G

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High Degree Witness

• High degree Witness
• Center Set W
• Clusters C1, .., Ck
• No useful intercluster edge

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In any MST Max Degree (W) d (W k 1) /
W e
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Feasible Swaps

• Feasible swap (e,f,T)
• Tree edge e
• Non-tree edge f
• Unique cycle in T f contains e
• c(e) c(f)
• A feasible swap (e,f,T)
• Produces an equal cost tree
• Reduces degree of endpoints of e

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Augmenting Paths

• Algorithm
• Construct an augmenting path of feasible swaps

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Algorithm Outline

• Start with
• Center Set W0
• Clusters connected through W0
• In step i
• Find an augmenting path of feasible swaps to
  improve some v in Wi
• Failure Center Set and clusters form a high
  degree witness

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Conclusion

• Improved approximation for
• BDMST (Bounded Degree MST)
• MSTDB (MST with Degree Bounds)
• New Techniques
• Improved cost bounding techniques based on
  Edmonds 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

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• Questions?

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Reduce-Max-Degree

• Initialize
• Let
• Sd nodes with degree d or more
• Pick d
• Sd-1 (log n/log log n) Sd
• Center Set
• W0 Sd Sd-1
• Initial clusters
• Components when W0 is deleted from T

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Reduce-Max-Degree

• For each useful intercluster edge f
• Find feasible swap (e,f,T) that improves v 2 Wt
• Remove v from Wt
• Form new cluster with
• e
• f
• v
• children clusters of v

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Reduce-Max-Degree

• Termination Conditions
• Degree d vertex v removed from Wt
• Can find a sequence of swaps to improve v
• Number of degree d vertices decreases by one

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Reduce-Max-Degree

• Termination Conditions
• Degree d vertex v removed from Wt
• Can find a set of swaps which improve v
• Number of degree d vertices decreases by one
• 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)

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Obtaining a Witness

• Center Set Wt
• Wt Sd
• Clusters
• From deleting Wt (d-2)Wt
• Lost from merges Sd-1
• Total (d-2)Wt - Sd-1
• Witness Quality
• At least (d-2) - O(log n/log log n)

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Summary

• Improved approximation for
• BDMST (Bounded Degree MST)
• MSTDB (MST with Degree Bounds)
• New Techniques
• Improved cost bounding techniques based on
  Edmonds 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

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A Better Algorithm

• Suppose given DH, we can find an MST with
• Max Degree 5DH 2
• Or show there is no MST with max degree DH
• BDMST Guarantees
• deg(v) 10(1?)D O(1)
• cost(T) (11/?) OPT
• MSTDB Algorithm uses Push-Relabel

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Push RelabelG85, 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

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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

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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) lt d-1 (d 1 deg(v)) units deficit

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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

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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

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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

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Future Directions

• Bounded Degree MSTs
• Improve guarantees to
• deg(v) B O(1)
• cost(T) OPT
• Embedding simple metrics to l1 with low distortion

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• Questions?

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Picking the Right Tree

• Proof
• We can show that
• C(TD) ?v ?Dv(degTD(v) D) OPTD
• If degTD(v) D when ?Dv gt 0
• C(TD) OPTD

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Picking the Right Tree

• KR00

• Our Work

• F93 Max Degree
• bB logb n

• Max Degree
• B O(log n/log log n)

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

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Picking the Right Tree

• KR00

• Our Work

• Guarantees
• Max degree
• B O(log n/log log n)
• Max Cost
• OPT
• Running Time
• Quasipolynomial

• Guarantees For ? gt 0
• Max degree
• (1 ?)bB logb n
• Max Cost
• (1 1/?) OPT
• Running time
• Polynomial

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• Augmenting paths and witnesses MSTs with Degree
  Bounds

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LP Formulation

• OPT Dual

max?v
minT (C(T) ?v ?v(degT(v)
- B)
MST in cost function cuv ?u ?v

• Properties
• Max degree B
• For all v such that ?v gt 0
• deg(v) B

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LP Formulation

• Primal
• min ?e ce xe
• ?e 2 ?(v) xe B
• x 2 SPG
• Dual

max?v
minT (C(T) ?v ?v(degT(v)
- B)
MST in cost function cuv ?u ?v
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LP Formulation

• Primal
• min ?e ce xe
• ?e 2 ?(v) xe B
• x 2 SPG
• Dual

max?v
minT (C(T) ?v ?v(degT(v)
- B)
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MSTDB Algorithm Outline

• Start with arbitrary MST T0
• In Phase i
• dmax max degree (Ti-1)
• dmin min degree (L)
• SH all nodes of degree dmax O(log n/log log
  n) or more
• SL all nodes in L of degree dmin O(log n/log
  log n) or less
• Try
• Improve a node in SH or SL
• Success New tree Ti
• Failure Witness for DH dmax O(log n/log log
  n) and DL dmin O(log n/log log n)

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Low Degree Witness

• Proof
• Any MST on W, C1, .., Ck has (W k 1) edges
• At most (2W k 2) endpoints can be in W
• Average degree of W
• b (2W k 2)/ W c

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