Approximating BuyatBulk and ShallowLight kSteiner Trees

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Approximating Buy-at-Bulk and Shallow-Light
k-Steiner Trees

• Mohammad T. Hajiaghayi (CMU)
• Guy Kortsarz (Rutgers)
• Mohammad R. Salavatipour (U. Alberta)
• Presented by
• Zeev Nutov

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Definition of Buy-at-Bulk k-Steiner Tree

• Given an undirected graph G(V,E), terminal set T?
  V, a root s?T, and integer k?T.
• Given two cost functions on the edges
• Buy cost
• Rent cost
• Goal find a subtree H spanning at least k
  terminals including root s minimizing
• where

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Motivation

• Network design problems with two cost functions
  have many applications, e.g. in bandwidth
  reservation when we have economies of scale
• Example capacity on a link can be purchased at
  discrete units
• with costs
• where

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Motivation (contd)

• So if you buy at bulk you save
• More generally, we have a concave function
  where f(b) is the minimum cost of cables with
  bandwidth b.

Question satisfy bandwidth for a set of demands
by installing sufficient capacities at minimum
cost
cost
bandwidth
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Equivalent Cost Measure

• Equivalent model cost distance
• There are a set of pairs
  to be connected
• For each possible cable connection e we can
• Buy it at b(e) and have unlimited bandwidth
• Rent it at r(e) and pay for each unit of flow
• A feasible solution buy and/or rent some edges
  to connect every si to ti.
• Goal minimize the total cost

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If this edge is bought its contribution to total
cost is 14.
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14
If this edge is rented, its contribution to total
cost is 2x36
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Total cost is where f(e) is the number of
paths going through e.
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Equivalent Cost Measure (contd)

• If E is the set of edges of the solution, the
  cost is
• where is the shortest
  path in
• We can think of as the start-up cost
  and
• as the per use cost (length).

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Special Cases

• If all sis (sources) are equal we have the
  single-source case (SS-BB)

Single-source

• If the cost and length functions on the edges
  are all the same, i.e. each edge e has cost
  clf(e) for constants c, l, we have the uniform
  case.

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12
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21
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Known Results for Buy-at-Bulk Problems

• Formally introduced by Salman et al. SCRS97
• O(log n) approximation for the uniform case
• AA97, Bartal98, FRT03
• O(log n) approx for the single-sink case
  MMP00
• Hardness of O(log log n) for the single-sink
  case CGNS05 and O(log1/2-? n) in general
  Andrews04, unless NP? ZPTIME(npolylog(n))
• Constant approx for several special cases
  AKR91,GW95,KM00,KGR02,KGPR02,GKR03
• Recently we gave an O(log4 n) approximation for
  the multicommodity case HKS06, CHKS06 .

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Shallow-Light k-Steiner Trees

• Instances are similar to BB k-Steiner tree
• an undirected graph G(V,E),
• terminals T? V,
• cost function,
• length function,
• a bound D and a parameter k ? T
• Find a tree spanning k terminals with minimum
  b-cost whose diameter under r-cost is at most D
  (assuming such a tree exists)
• (?,?)-bicriteria approx cost at most ?.opt and
  diameter is at most ?.D where opt is the cost of
  optimum solution with diameter bound D

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Our Results

• Theorem 1 Given an instance of shallow-light
  k-Steiner tree with bound D, we find a
  (k/8)-Steiner tree with diameter O(log n.D) and
  cost O(log3n.opt).
• Corollary we get an (O(log2 n),O(log3
  n))-bicriteria approx for shallow-light k-Steiner
  tree
• Theorem 2 There is an O(log4 n)-approximation
  for buy-at-bulk k-Steiner tree.
• Note
• BB k-Steiner generalizes k-MST and k-Steiner
  (when r0).
• Shallow-light k-Steiner generalizes shallow-light
  Steiner (when kT ) and k-MST (when D1).

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How to Reduce BB to Shallow-Light

• Let G be an instance of BB and assume we know the
  value of OPT (e.g. by guessing).
• Lemma If there is an (?,?)-bicriteria algorithm
  A for shallow-light k-Steiner that finds a
  (k/8)-Steiner tree, then there is an O((?? ) log
  n) approx for BB k-Steiner.
• Proof
• First, we can ignore every vertex with r-distance
  gtOPT from the root.
• Then we run the following algorithm.

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How to Reduce BB to Shallow-Light (contd)

• While kgt0 repeat the following
• Run the (?,?)-approx alg A for (k/2)-Steiner
  tree with diameter bound D4OPT/k
• Decrease k by the number of terminals covered in
  the new solution mark all these terminals as
  Steiner nodes goto 1
• The union of the solutions found is returned.
• Consider some iteration and let k be the number
  of unspanned terminals and H be an optimal
  solution for BB k-Steiner.

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How to Reduce BB to Shallow-Light (contd)

• Iteratively remove leaves (terminals)
• with r-distance gt 2OPT/k from H.
• We delete at most k/2 terminals and r-diameter
  is at most 4.OPT/k
• Using alg A we find a (k/16)-Steiner tree with
  diameter bound 4?.OPT/k. This adds at most
  k.?.2OPT/k2?.OPT to the rent cost buy cost is
  at most ?.OPT
• So we have covered a constant fraction of k at
  cost at most O((??).OPT).
• A standard set-cover analysis shows the total
  cost is in O((??).OPT.log n).

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Overview of Algorithm for Shallow-Light k-Steiner

• First we compute a completion graph Gc of G
• for every pair u,v?V, compute
  (approximately) the minimum b-cost u,v-path with
  r-cost at most 2D. It is easy to show
• Lemma if there is a bicriteria solution of cost
  X and diameter Y in Gc then we can find a
  solution of cost X and diameter Y in G.
• So it is enough to work with Gc.
• Also, we can easily transform the un-rooted case
  and the rooted case to each other.

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Overview of Algorithm (contd)

• We maintain a collection of trees
• At the beginning every terminal is a
• tree of one node
• We design a test that can fail or succeed
• If the test succeds two trees are merged
• Else some terminals are temporarily deleted

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Overview of Algorithm (contd)

• We maintain a collection of trees partition
• According to their number of terminals

1 to 2 terminals
3 to 4 terminals
p to 2p terminals
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The Test

• Pick a cluster of p to 2p terminals that
  contains many roots
• Every root is a terminal
• A terminals is a TRUE terminal if belongs
• to the optimum
• The test does the collection of roots contain
  many terminals?

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The Main Argument

• If the test succeeds then two trees
• are contracted together at a low price
• If it fails all roots in the cluster are removed
• We loose many terminals
• But only few true terminals
• Hence eventually a tree will reach size k/8

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Conclusion and Open Problems

• We obtain O(log4 n) approximation algorithm for
  buy-at-bulk k-steiner trees. The current lower
  bound is only O(log log n).
• Main open problem Can we improve the upper bound
  significantly or at least the lower bound to
  O(log n)?

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• Thank you.