Approximation Algorithms Approximation Algorithms for Non-Uniform Buy-at- for Non-Uniform Buy-at- Bulk Network Design Bulk Network Design Problems Problems MohammadTaghi Hajiaghayi MohammadTaghi Hajiaghayi Carnegie Mellon University Carnegie Mellon University Joint work with Joint work with Chandra Chekuri Chandra Chekuri (UIUC) (UIUC) Guy Kortsarz Guy Kortsarz (Rutgers, Camden) (Rutgers, Camden) Mohammad R. Salavatipour Mohammad R. Salavatipour ( ( University of University of Alberta) Alberta)
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Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design Problems
Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design Problems. MohammadTaghi Hajiaghayi Carnegie Mellon University Joint work with Chandra Chekuri (UIUC) Guy Kortsarz (Rutgers, Camden) Mohammad R. Salavatipour ( University of Alberta). Motivation. - PowerPoint PPT Presentation
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Approximation Algorithms for Approximation Algorithms for Non-Uniform Buy-at-Bulk Non-Uniform Buy-at-Bulk Network Design ProblemsNetwork Design Problems
MohammadTaghi HajiaghayiMohammadTaghi HajiaghayiCarnegie Mellon UniversityCarnegie Mellon University
Mohammad R. Salavatipour Mohammad R. Salavatipour ((University of Alberta)University of Alberta)
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MotivationMotivation
Suppose we are given a network and some Suppose we are given a network and some nodes have to be connected by cablesnodes have to be connected by cables
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Each cable has a cost (installation or cost of usage)
Question: Install cables satisfying demands at minimum cost
This is the well-studied Steiner forest problem and is NP-hard
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Motivation (cont’d)Motivation (cont’d)
Consider buying bandwidth to meet demands Consider buying bandwidth to meet demands between pairs of nodes.between pairs of nodes.The cost of buying bandwidth satisfy The cost of buying bandwidth satisfy economies of scaleeconomies of scaleThe capacity on a link can be purchased at The capacity on a link can be purchased at discrete units:discrete units:
Costs will be: Costs will be:
WhereWhere
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So if you buy at bulk you saveSo if you buy at bulk you saveMore generally, we have a non-decreasing More generally, we have a non-decreasing monotone concave (or even sub-additive) functionmonotone concave (or even sub-additive) function
where where f f ((bb)) is the minimum cost of is the minimum cost of cables with bandwidth cables with bandwidth bb..
Motivation (cont’d)Motivation (cont’d)
bandwidth
cost
Question: Given a set of bandwidth demands between nodes, install sufficient capacities at minimum total cost
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Motivation (cont’d)Motivation (cont’d)
The previous problem is equivalent to the following The previous problem is equivalent to the following problem:problem:
There are a set of pairs There are a set of pairs
to be connectedto be connected
For each possible cable connection For each possible cable connection ee we can: we can:
Buy it at Buy it at bb((ee)): : and have unlimited useand have unlimited use
Rent it at Rent it at rr((ee)): : and pay for each unit of flowand pay for each unit of flow A feasible solutionA feasible solution: buy and/or rent some edges to connect : buy and/or rent some edges to connect
every every ssii to to ttii. .
Goal: Goal: minimize the total cost minimize the total cost
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Motivation (cont’d)Motivation (cont’d)
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If this edge is bought its contribution to total cost is 14.
If this edge is rented, its contribution to total cost
is 2x3=6
Total cost is:
where f(e) is the number of paths going over e.
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These problems are also known as cost-distance These problems are also known as cost-distance problems:problems:
cost functioncost function
length function length function
Also a set of pairs of nodes each with a Also a set of pairs of nodes each with a demand for every demand for every i i
Feasible solution: a set s.t. all pairs Feasible solution: a set s.t. all pairs
are connected in are connected in
Cost-DistanceCost-Distance
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Cost-Distance (cont’d)Cost-Distance (cont’d)
The cost of the solution is:The cost of the solution is:
where is the shortest path in where is the shortest path in
The cost is the start-up cost andThe cost is the start-up cost and
is the per-use cost (length).is the per-use cost (length).
Goal:Goal: minimize total cost. minimize total cost.
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Multicommodity Buy At BulkMulticommodity Buy At Bulk
Note that the solution Note that the solution may have cyclesmay have cycles
Special CasesSpecial CasesIf all If all ssi i (sources) are equal we have the (sources) are equal we have the single-source case single-source case
(SS-BB)(SS-BB)
If the cost and length functions on the edges are all the same, i.e. each edge e has cost c + l f(e) for constants c,l : Uniform-case
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Single-source
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Previous WorkPrevious Work
Formally introduced by F. S. Salman, J. Cheriyan, R. Ravi Formally introduced by F. S. Salman, J. Cheriyan, R. Ravi and S. Subramanian, 1997and S. Subramanian, 1997
OO(log(log n n)) approximation for the uniform case, i.e. each approximation for the uniform case, i.e. each edge edge ee has cost has cost c+lc+lff((ee)) for some fixed constants for some fixed constants c, lc, l (B. (B. Awerbuch and Y. Azar, 1997;Awerbuch and Y. Azar, 1997; YY. . Bartal, 1998)Bartal, 1998)
OO(log(log n n)) randomized approximation for the single-sink randomized approximation for the single-sink case: Acase: A. . Meyerson, K. Munagala and S. Plotkin, 2000Meyerson, K. Munagala and S. Plotkin, 2000
OO(log(log n n) ) deterministic approximation for the single-sink deterministic approximation for the single-sink case: case: C. Chekuri, S. Khanna and S. Naor, 2001C. Chekuri, S. Khanna and S. Naor, 2001
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Hardness Results for Buy-at-Bulk Hardness Results for Buy-at-Bulk ProblemsProblems
Hardness of Hardness of ΩΩ(log log(log log n n)) for the single- for the single- sink case sink case JJ. Chuzhoy, A. Gupta, J. Naor . Chuzhoy, A. Gupta, J. Naor and A. Sinha, 2005and A. Sinha, 2005
ΩΩ(log(log1/2-1/2- n n)) in general M. Andrews 2004, in general M. Andrews 2004, unless unless NPNP ZPTIME(n ZPTIME(npolylog(n)polylog(n)))
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Algorithms for Special Cases Algorithms for Special Cases
Steiner ForestSteiner Forest
A. Agrawal, P. Klein and R. Ravi, 1991A. Agrawal, P. Klein and R. Ravi, 1991M. X. Goemans and D. P. Williamson, 1995 M. X. Goemans and D. P. Williamson, 1995
Single sourceSingle source
S. Guha, A. Meyerson and K. Munagala , 2001S. Guha, A. Meyerson and K. Munagala , 2001K. Talwar, 2002K. Talwar, 2002A. Gupta, A. Kumar and T. Roughgarden, 2002A. Gupta, A. Kumar and T. Roughgarden, 2002A. Goel and D. Estrin, 2003A. Goel and D. Estrin, 2003
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Multicommodity Buy at BulkMulticommodity Buy at Bulk
Multicommodity Uniform Case:Multicommodity Uniform Case:Y. Azar and B. Awerbuch, 1997Y. Azar and B. Awerbuch, 1997
Y. Bartal,1998Y. Bartal,1998
A. Gupta, A. Kumar, M. Pal and T. Roughgarden, A. Gupta, A. Kumar, M. Pal and T. Roughgarden, 20032003
The only known approximation for the general caseThe only known approximation for the general case
M. Charikar, A. Karagiozova, 2005. The ratio is:M. Charikar, A. Karagiozova, 2005. The ratio is:
exp( exp( OO(( log (( log DD log log log log D D ))1/2 1/2 ))))
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Our Main ResultOur Main Result
Theorem: Theorem: If If hh is the number of pairs of is the number of pairs of ssii,t,tii then there is a polytime algorithm with then there is a polytime algorithm with approximation ratio approximation ratio OO(log(log44 h h))..
For simplicity we focus on the unit-demand For simplicity we focus on the unit-demand case (i.e. case (i.e. ddii==11 for all for all i’i’s) and we present s) and we present
OO(log(log55nn loglogloglog n n))..
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Overview of the AlgorithmOverview of the Algorithm
The algorithm iteratively finds a partial The algorithm iteratively finds a partial solution connecting some of the residual solution connecting some of the residual pairspairs
The new pairs are then removed from the set; The new pairs are then removed from the set; repeat until all pairs are connected (routed)repeat until all pairs are connected (routed)
Density of a partial solution = Density of a partial solution = cost of the partial solutioncost of the partial solution # of new pairs routed# of new pairs routed
The algorithm tries to find low density partial The algorithm tries to find low density partial solution at each iterationsolution at each iteration
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Overview of the Algorithm (cont’d)Overview of the Algorithm (cont’d)
The density of each partial solution is at mostThe density of each partial solution is at most
ÕÕ(log(log44 nn) ) (OPT / (OPT / hh'')) where where OPTOPT is the cost of is the cost of optimum solution and optimum solution and hh'' is the number of is the number of unrouted pairsunrouted pairs
A simple analysis (like for set cover) shows:A simple analysis (like for set cover) shows:
How to compute a low-density partial solution?How to compute a low-density partial solution?
Prove the existence of low-density one with a very Prove the existence of low-density one with a very specific structure: specific structure: junction-treejunction-tree
Junction-tree:Junction-tree: given a set given a set PP of pairs, tree of pairs, tree TT rooted at rooted at rr is a junction tree if is a junction tree if
It contains all pairs of It contains all pairs of PP
For every pair For every pair ssii,t,tii P P the the
path connecting them path connecting them
in in TT goes through goes through rr
r
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Structure of the Optimum (cont’d)Structure of the Optimum (cont’d)
So the pairs in a junction tree connect via the rootSo the pairs in a junction tree connect via the root
We show there is always a partial solution with low We show there is always a partial solution with low density that is a junction treedensity that is a junction tree
Observation:Observation: If we know the pairs participating in a If we know the pairs participating in a junction-tree it reduces to the single-source BB junction-tree it reduces to the single-source BB problemproblem r
Then we could use the Then we could use the
OO(log(log n n)) approximation approximation
of of [MMP’00][MMP’00]
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Summary of the AlgorithmSummary of the Algorithm
So there are two main ingredients in the proofSo there are two main ingredients in the proof
Theorem 2:Theorem 2: There is always a partial solution that is a There is always a partial solution that is a
junction tree with density junction tree with density ÕÕ (log(log22 nn) ) (OPT / (OPT / hh''))
Theorem 3:Theorem 3: There is an There is an O O (log(log22 nn)) approximation for the approximation for the
problem of finding lowest density junction tree (this is low problem of finding lowest density junction tree (this is low
density SS-BB).density SS-BB).
Corollary:Corollary: We can find a partial solution with density We can find a partial solution with density
ÕÕ (log(log44 nn) ) (OPT / (OPT / hh''))
This implies an approximation This implies an approximation ÕÕ (log(log55 nn)) for MC-BB. for MC-BB.
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More Details of the Proof of Theorem 2: More Details of the Proof of Theorem 2:
We want to show there is a partial solution that is a junction tree We want to show there is a partial solution that is a junction tree
with density with density ÕÕ (log(log22 nn) ) (OPT / (OPT / hh''))
Consider an optimum solution OPT.Consider an optimum solution OPT.
Let Let E*E* be the edge set of OPT, be the edge set of OPT, OPTOPTcc be its costbe its cost and and
OPTOPTll be be its length. its length.
By the result of By the result of Elkin, Emek, Spielman and Tang 2005Elkin, Emek, Spielman and Tang 2005 on on
probabilistic distribution on spanning trees and by loosing a probabilistic distribution on spanning trees and by loosing a
factor factor ÕÕ (log(log22 nn)) on length, we can assume that on length, we can assume that E* E* is a forest is a forest T T
(WLOG we assume (WLOG we assume TT is connected). is connected).
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More Details of the Proof of Theorem 2: More Details of the Proof of Theorem 2:
From From TT we obtain a collection of rooted subtrees we obtain a collection of rooted subtrees TT11,…,T,…,Taa
such thatsuch that
any edge e of any edge e of TT is in at most is in at most O(O(loglog h) h) of the subtrees of the subtrees
For every pair there is exactly one index For every pair there is exactly one index ii such that both vertices such that both vertices
are in are in TTii; further the root of ; further the root of TTii is their least common ancestor is their least common ancestor
The total cost of the junction trees is at most The total cost of the junction trees is at most