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A Push-Relabel Algorithm for Approximating
Degree-Bounded Minimum Spanning Trees

• Kamalika Chaudhuri
• Satish Rao U.C. Berkeley
• Sam Riesenfeld
• Kunal Talwar Microsoft Research

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bounded-degree MST (BDMST) problem

• Given graph G(V,E), edge costs ce,
• and degree bound Bgt1
• Find a spanning tree such that c(T) is
  minimized,
• subject to deg(T) B.

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bounded-degree MST (BDMST) problem

• Given graph G(V,E), edge costs ce,
• and degree bound Bgt1
• Find a spanning tree such that c(T) is
  minimized,
• subject to deg(T) B.

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cost 8
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reduction to Traveling Salesman Path

• For B2, BDMST equivalent to TSPP
• ) BDMST is NP-Complete.
• Approximation algorithm is unlikely unless we
  relax degree bounds.
• (Edge costs ce may not satisfy triangle
  inequality.)

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minimum-degree MST (MDMST) problem

• Consider a variant of this cost-degree trade-off
• amongst all (true) MSTs,
• find one that minimizes the degree.
• (Note trivial when there is a unique MST.)

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minimum-degree MST (MDMST) problem

• Consider a variant of this cost-degree trade-off
• amongst all (true) MSTs,
• find one that minimizes the degree.
• (Note trivial when the MST is unique.)

degree 4
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previous work

• Fürer, Raghavachari 94
• Unweighted MDMST/BDMST 1 approximation
• Fischer 93
• MDMST Local Search gives 2BOPTlog2n
• Könemann, Ravi 00, 03
• Above algorithm Lagrangean LP relaxation )
• A tree with cost at most 2cOPT(B)
• and degree at most 2(2Blog2n)
• i.e. a (2, 4B2log2n)-bicriteria approximation

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previous work

• Chaudhuri, Rao, R., Talwar 05
• New MDMST algorithm using augmenting paths
  (log2n/loglog2n) approximation in
  quasi-polynomial time.
• Improved cost-bounding techniques for BDMST
• (1, B(log2n/loglog2n))-approximation

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this talk

• New MDMST algorithm
• (2BOPT O(vBOPT))-approximation
• Using it for BDMST
• (2, 4BOPT O(vBOPT))-approximation
• Inspired by Goldbergs push-relabel algorithm for
  max-flow Goldberg87.

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other recent work

• Ravi, Singh in ICALP 06
• MDMST k approximation
• where k distinct cost classes.
• Goemans 06
• 2 approximation, upcoming FOCS 06

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the push-relabel framework for MDMST

• Start with an arbitrary MST T
• Repeat
• Try to improve the max degree of T
• Either decrease max degree by 1
• Or find a proof that BOPT must be large, and
  stop.

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valid swaps

• Valid swap (e, e)
• Add an edge e to the MST delete a same-cost
  edge e from the cycle.

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labeling and feasibility

• Each node has a label.
• Each edge has a label label of higher endpoint.
• A valid swap (e, e) is called feasible if
• label(e) label(e) 1
• Lemma (Surprisingly) valid swaps are always
  feasible.

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lower bounding BOPT

• A center set W
• A partition of V-W into clusters C1,,Ck
• No Ci-Cj edges exist in any MST
• (In any MST, all improvements to a node in W
  increase degree of another node in W.)
• Average degree of W must be k/W in any MST.
• We call W a witness.

W
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the push-relabel algorithm

• Try to improve the max degree of T
• Give each node a label (0 to begin).
• Give excess of 1 to each node of max degree.
• Nodes with excess can push it onto
  lower-labeled nodes.
• If a node with excess has feasible swap pushing
  excess down
• Execute swap. Adjust excesses.
• Else raise labels of some nodes with excess.
• Either max degree decreases by 1, done
• Or we raise some label to log2n, then find a
  witness.

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example
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using feasibility to get a lower bound

• If the push-relabel algorithm stops with a gap in
  the labeling
• Let W be the set of nodes above the gap.
• Feasibility ) no valid swaps for W
• ) W is a witness
• W is a good witness if there are many components
  below the gap.
• In fact, a sparse level suffices.

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the lower bound

• Theorem The push-relabel MDMST algorithm finds
  a witness certifying that
• deg(T) 2BOPT O(vBOPT).
• Proof Uses careful charging argument bounding
  the average degree of W in T.
• (Cascades of falling excess dont hurt too
  much.)

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Thank you

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MDMST (degree 4, cost 8) backup slide
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MST (degree 5, cost 8) backup slide
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