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Approximation Some Network Design Problems With Node Costs

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### Approximation Some Network Design Problems With Node Costs

Guy Kortsarz

Rutgers University, Camden, NJ

Joint work with

Zeev Nutov

The Open University, Israel

The problems Studied. Only Node Costs.

Multicommodity Buy At Bulk with Node

Costs:

Input: an undirected graph and for every s,tVV,

a demand dst,.

Every vertex v has subadditive cost function g(v).

Remark: this represents different routers type per

vertex and economies of scale. Subadditive is a

discrete relaxation of concave function on R

The requirement and objective 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

Cost-Distance

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

The Minimum Multicommodity Cost-Distance problem:

Input:A graph G(V,E)Cost functionc:V R

length function l: V R+, and for everys,tVV

a demand dst

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

Cost-Distance (cont’d)

- The cost of the solution is:

c(V’)+ s,t dst distV’(s,t)

- Where distV’(s,t) is the weighted distance between s and t in the graph induced by V’
- So you have fixed cost (like in Steiner forest) paid for every vV’ . This is called the FIX COST.
- But every d demand units that go via

v induce a cost of l(v)·d. This is called the INCREMENTAL COST.

Our First result

- The previously best known approximation for

MBB and Cots-Distance was O(log4 n).

- By Chekuri et al.
- We give an O(log3n) polynomial time approximation ratio for the case the demands are polynomial in n
- Remark: For exponential demand the best known is still O(log4 n)

Our Second probelm

- The Tree Covering (MaxTC) Problem:

Given a graph G with vertex costs vertex profits and budget bound B, find a maximum profit subtree T Gof budget at most B

- Previous work:

1) First algorithm: S. Guha, A. Moss, S. Naor,

Y. Rabani and B. Schieber.

2B cost, opt/O(log2n) profit

2) Improvement: Moss and Rabani.

2B cost, opt/O(log n) profit

- Conjectured by Moss and Rabani to have O(1) approximation ratio

Our Second Result

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

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

Our Third Problem

Shallow-Light trees with node costs:

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

length l(v) and a cost bound c and

diameter bound L

Output: A subtree with cost c and diameter L

Our Third Result

We find a subtree T with cost

O(log n) c and diameter O(log2n) L.

Remark:M. Marathe, R. Ravi,

Ravi Sundaram , S. S. Ravi,

Daniel J. Rosenkrantz, and

Harry B. Hunt III, gave a similar

algorithm for edge weights

Their ratio is O(log n,log n)

Motivation for MBB

- Consider buying routers to meet demands between pairs of nodes.
- The cost of buying routers satisfy economies of scale
- The capacity on a node can be purchased at discrete units:

Costs will be:

Where

Motivation (cont’d)

- So if you buy at bulk you save
- More generally, we have a non-decreasing monotone concave functiongv: RR for every vwhere gv(b) is the minimum cost of a router/switcher with bandwidth b.

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

cost

bandwidth

Ilustration of the cost-distance variant

t1

1,4

t2

1,6

2,3

2,5

w

1,3

2,9

2,8

2,3

u

v

s1

2,4

1,3

s2

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

Overview of the Algorithm for cost-distance

- The algorithm iteratively finds a partial solution connecting some of the residual pairs
- The new pairs are then removed from the set; repeat until all pairs are connected (routed)
- Density of a partial solution =

cost of the partial solution

# of new pairs routed

- The algorithm tries to find low density partial solution at each iteration

Junction trees

- A tree is a junction tree if it can be rooted by a node r so that all (unique) s,t paths go via the rootr
- For polynomial demands the density penalty in cost for best junction tree is O(1) and in length O(log n)
- For exponential demands O(log n) payment in both measures
- Given that we can find an approximate density solution by so called density LP’s

How do we improve?

- Chandra et al proved:

There is a junction tree with cost O(optc/h)

However the diameter is O(log n)· optl/h

Note that there is an O(log n) advantage

for the cost in this lemma.

- There are what we call density LP that will induce a penalty of O(log2n) in the density of the actual tree found
- As stated, O(log3 n) density for length and one more

O(log n) for set cover type payment, O(log4n)

Saving a log n

- We defined a new LP in which the incremental cost is not in the objective function
- Instead all paths used can have length at most A·optl/h with A universal constant
- Solve this LP by duality
- Intuitively the returned set is smaller by a

log n factor than the opt set. BUT THIS HAS NO AFFECT AT LENGTH DENSITY

Summary

- Cost: Looses nothing from the junction tree lemma, looses O(log2 n) from density LP and looses O(log n) from set cover analysis
- Length: Looses O(log n) from junction tree lemma, only one O(log n) from density LP and one O(log n) from set-cover analysis.
- In both cases O(log3n) times the cost and length optima

Open Problems

- Our guess is that MaxCT should have

(log n) lower bound (currently only

(log log n) )

- Our guess is that MBB and Cost-Distance should have O(log2 n) upper (and lower?) bound. Even with exponential demands.
- Finally, we guess that shallow-light trees with nodes cost can not have

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

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