Approximation Some Network Approximation Some Network Design Problems With Node CostsDesign Problems With Node Costs

Guy KortsarzGuy KortsarzRutgers University, Camden, NJRutgers University, Camden, NJ

Joint work withJoint work with

Zeev Nutov Zeev Nutov The Open University, IsraelThe Open University, Israel

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The problems Studied. Only Node The problems Studied. Only Node Costs.Costs.

Multicommodity Buy At Bulk with NodeMulticommodity Buy At Bulk with Node Costs:Costs:

InputInput: an undirected graph and : an undirected graph and for every s,tVV, a demand dst,. Every vertex v has subadditivesubadditive cost function g(v) cost function g(v). Remark: this represents different routers type pervertex and economies of scale. Subadditive is adiscrete relaxation of concave function on R

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The requirement and objective The requirement and objective function:function:

Required: define a flow of ds,t between every s,t (no capacity bounds)

Let f(v) be the total flow going via v. Objective function: Minimize:

v gv(f(v))

Call this problem MBB

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The following problem is equivalent to approximate The following problem is equivalent to approximate within ratio 2 to the MBB problems:within ratio 2 to the MBB problems:

The Minimum Multicommodity Cost-Distance problem:The Minimum Multicommodity Cost-Distance problem:

Input:Input: A graph A graph G(V,E)G(V,E) Cost functionCost function c:Vc:V R R

length function length function l: V l: V R R++, and , and for every s,tVV

a demand dst

Required:Required: A feasible solution is a subset V’V such that for every s,t of demand larger than 0, s and t

have finite distance in graph induced by V’

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:

c(V’)+ c(V’)+ s,ts,t d dst st distdistV’V’(s,t)(s,t)

Where Where distdistV’V’(s,t)(s,t) is the weighted distance is the weighted distance

between between ss and and tt in the graph induced by in the graph induced by V’V’So you have fixed cost (like in Steiner forest) paid for So you have fixed cost (like in Steiner forest) paid for every every vvV’V’ . This is called the . This is called the FIX COSTFIX COST..

But every But every dd demand units that go via demand units that go via

vv induce a cost of induce a cost of ll((vv))··dd. . This is called the This is called the INCREMENTAL COST.INCREMENTAL COST.

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Our First resultOur First result

The previously best known approximation for The previously best known approximation for

MBB and Cots-Distance was MBB and Cots-Distance was O(O(loglog44 n). n). By Chekuri et al.By Chekuri et al. We give an We give an O(O(loglog33n)n) polynomial time polynomial time

approximation ratio for the case the demands approximation ratio for the case the demands are polynomial in are polynomial in nn

Remark: For exponential demand the best Remark: For exponential demand the best known is still known is still O(O(loglog44 n) n)

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Our Second probelmOur Second probelm

The Tree Covering (MaxTC) Problem:The Tree Covering (MaxTC) Problem: Given a graph Given a graph GG with vertex costs vertex profits and with vertex costs vertex profits and

budget bound budget bound BB, find a maximum profit subtree , find a maximum profit subtree TT G G of budget at most of budget at most BBPrevious work:Previous work:

1) First algorithm: S. Guha, A. Moss, S. Naor, 1) First algorithm: S. Guha, A. Moss, S. Naor, Y. Rabani and B. Schieber. Y. Rabani and B. Schieber. 2B2B cost, cost, optopt//OO(log(log22 nn)) profit profit 2) Improvement: Moss and Rabani. 2) Improvement: Moss and Rabani. 2B2B cost, cost, opt/Oopt/O(log(log n n)) profit profit

Conjectured by Moss and Rabani to have Conjectured by Moss and Rabani to have OO(1)(1) approximation ratioapproximation ratio

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Our Second ResultOur Second Result

Unless Unless NPNP admits a quasi-polynomial solution admits a quasi-polynomial solution MaxCT admits no MaxCT admits no (loglog n)(loglog n) ratio ratio approximation approximation even if the solution is allowed to violate the budget even if the solution is allowed to violate the budget by a universal constant by a universal constant (as Moss and Rabani with (as Moss and Rabani with =2=2)) Disproves the conjecture by Moss and Rabani.Disproves the conjecture by Moss and Rabani.Also Unless Also Unless P=NPP=NP, no constant approximation exists , no constant approximation exists

for any universal constant for any universal constant cc even if the solution is even if the solution is allowed to violate the budget within any universal allowed to violate the budget within any universal constant constant

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Our Third ProblemOur Third Problem

Shallow-Light trees with node costs:Shallow-Light trees with node costs:

InputInput: A graph G(V,E) with costs : A graph G(V,E) with costs c(v)c(v) and and

length length l(v) l(v) and a cost bound and a cost bound cc and and

diameter bound diameter bound LL

Output: Output: A subtree with cost A subtree with cost cc and diameter and diameter LL

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Our Third ResultOur Third Result

We find a subtree T with cost We find a subtree T with cost O(O(loglog n) n) cc and diameter and diameter O(O(loglog22 n) Ln) L..Remark:Remark: M. Marathe, R. Ravi, M. Marathe, R. Ravi, Ravi Sundaram , S. S. Ravi, Ravi Sundaram , S. S. Ravi, Daniel J. Rosenkrantz, and Daniel J. Rosenkrantz, and Harry B. Hunt III, gave a similar Harry B. Hunt III, gave a similar algorithm for edge weights algorithm for edge weights Their ratio is Their ratio is O(O(loglog n, n,loglog n) n)

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Motivation for MBBMotivation for MBB

Consider buying routers to meet demands Consider buying routers to meet demands between pairs of nodes.between pairs of nodes.The cost of buying routers satisfy economies The cost of buying routers satisfy economies of scaleof scaleThe capacity on a node can be purchased at The capacity on a node 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 monotone More generally, we have a non-decreasing monotone concave functionconcave function ggvv: R: RRR for every for every vv where where ggvv((bb)) is is the minimum cost of a router/switcher with the minimum cost of a router/switcher with bandwidth bandwidth bb..

Motivation (cont’d)Motivation (cont’d)

bandwidth

cost

Question: Given a set of bandwidth demands between nodes, install sufficient capacities at minimum cost. The cost per v is non-decreasing concave

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Ilustration of the cost-distance variantIlustration of the cost-distance variant

s2

t1

s1

1,6

2,9

t2

2,5

1,3

2,31,3

1,4

v u

C(V')= c(s1)+c(u)+c(v)+c(w)+c(t1)+c(s2)+c(t2)=38

l(V')=2·l(u)+2·l(v)+l(w)+l(s1)+l(t1)+l(s2)+l(t2)=13

w

2,8

2,4

2,3

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Overview of the Algorithm for cost-distanceOverview of the Algorithm for cost-distance

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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Junction treesJunction trees

A tree is a junction tree if it can be rooted by a node A tree is a junction tree if it can be rooted by a node rr so that all (unique) so that all (unique) s,ts,t paths paths go via the rootgo via the root rr

For polynomial demands the density penalty in cost For polynomial demands the density penalty in cost for best junction tree is for best junction tree is O(O(11)) and in length and in length O(O(loglog n) n)

For exponential demands For exponential demands O(log n)O(log n) payment in both payment in both measures measures

Given that we can find an approximate density Given that we can find an approximate density solution by so called solution by so called density LP’sdensity LP’s

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How do we improve?How do we improve?

Chandra et al proved:Chandra et al proved: There is a junction tree with cost There is a junction tree with cost O(optO(optcc/h) /h)

However the diameter is However the diameter is O(O(loglog n) n)·· opt optll/h/h

Note that there is an Note that there is an O(O(loglog n) n) advantage advantage for the cost in this lemma. for the cost in this lemma.

There are what we call There are what we call density LPdensity LP that will induce a penalty that will induce a penalty of of OO(log(log22nn) ) in the density of the actual tree foundin the density of the actual tree found

As stated, As stated, OO(log(log33 n n)) density for length and one more density for length and one more OO(log (log nn)) for set cover type payment, for set cover type payment, OO(log(log44 nn))

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Saving a Saving a loglog n n

We defined a new LP in which the incremental We defined a new LP in which the incremental cost is not in the objective functioncost is not in the objective function

Instead all paths used can have length at most Instead all paths used can have length at most AA··optoptll//hh with A universal constant with A universal constant

Solve this LP by dualitySolve this LP by duality Intuitively the returned set is smaller by a Intuitively the returned set is smaller by a

loglog n n factor than the factor than the optopt set. set. BUT THIS HAS BUT THIS HAS NO AFFECT AT LENGTH DENSITYNO AFFECT AT LENGTH DENSITY

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SummarySummary

Cost: Looses nothing from the junction tree Cost: Looses nothing from the junction tree lemma, looses lemma, looses O(O(loglog22 n) n) from density LP and from density LP and looses looses O(O(loglog n) n) from set cover analysis from set cover analysis

Length: Looses Length: Looses O(O(loglog n) n) from junction tree from junction tree lemma, only one lemma, only one OO(log (log nn)) from density LP and from density LP and one one OO(log (log nn)) from set-cover analysis. from set-cover analysis.

In both cases In both cases OO(log(log33 nn)) times the cost and times the cost and length optima length optima

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Open ProblemsOpen Problems

Our guess is that MaxCT should have Our guess is that MaxCT should have ((loglog n) n) lower bound (currently only lower bound (currently only ((log loglog log n) n) ) ) Our guess is that MBB and Cost-Distance Our guess is that MBB and Cost-Distance

should have should have O(O(loglog22 n) n) upper (and lower?) upper (and lower?) bound. Even with exponential demands.bound. Even with exponential demands.

Finally, we guess that shallow-light trees with Finally, we guess that shallow-light trees with nodes cost nodes cost can notcan not have have

(O((O(loglog n),O( n),O(loglog n)) n)) ratio. Proof anyone? ratio. Proof anyone?