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Approximating BuyatBulk and ShallowLight kSteiner Trees

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Title: Approximating BuyatBulk and ShallowLight kSteiner Trees

1
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

2
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

3
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

4
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
5
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

6
If this edge is bought its contribution to total
cost is 14.
10
14
If this edge is rented, its contribution to total
cost is 2x36
3
Total cost is where f(e) is the number of
paths going through e.
7
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).

8
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.

5
12
8
21
11
9
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 .

10
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

11
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).

12
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.

13
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.

14
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).

15
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.

16
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

17
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
18
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?

19
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

20
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)?