1. Approximation Some Network Design Problems With Node Costs Guy Kortsarz Rutgers University, Camden, NJ Joint work with Zeev Nutov The Open University, Israel

2. 2 The problems Studied. Only Node Costs. Multicommodity Buy At Bulk with Node Multicommodity Buy At Bulk with Node Costs: Costs: Input: an undirected graph and Input: an undirected graph and for every s,t  V  V, a demand d st,. subadditive cost function g(v) 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

3. 3 The requirement and objective function: Required: define a flow of d s,t between every s,t (no capacity bounds) Let f(v) be the total flow going via v. Objective function: Minimize :  v g v (f(v)) Call this problem MBB

4. 4 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 function c:V  R length function l: V  R +, and length function l: V  R +, and for every s,t  V  V a demand d st 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-Distance

5. 5 Cost-Distance (cont’d) The cost of the solution is:  c(V’)+  s,t d st dist V’ (s,t)  c(V’)+  s,t d st dist V’ (s,t) Where dist V’ (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 v  V’. 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. v induce a cost of l(v)·d. This is called the INCREMENTAL COST.

6. 6 Our First result  The previously best known approximation for MBB and Cots-Distance was O(log 4 n). MBB and Cots-Distance was O(log 4 n).  By Chekuri et al.  We give an O(log 3 n) polynomial time approximation ratio for the case the demands are polynomial in n  Remark: For exponential demand the best known is still O(log 4 n)

7. 7 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  G of budget at most B Given a graph G with vertex costs vertex profits and budget bound B, find a maximum profit subtree T  G of budget at most B 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. 2B cost, opt/O(log 2 n) profit 2B cost, opt/O(log 2 n) profit 2) Improvement: Moss and Rabani. 2) Improvement: Moss and Rabani. 2B cost, opt/O(log n) profit 2B cost, opt/O(log n) profit Conjectured by Moss and Rabani to have O(1) approximation ratio

8. 8 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) 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. 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  for any universal constant c even if the solution is allowed to violate the budget within any universal constant 

9. 9 Our Third Problem Shallow-Light trees with node costs: Input: A graph G(V,E) with costs c(v) and Input: A graph G(V,E) with costs c(v) and length l(v) and a cost bound c and length l(v) and a cost bound c and diameter bound L diameter bound L Output: A subtree with cost c and diameter L Output: A subtree with cost c and diameter L

10. 10 Our Third Result We find a subtree T with cost O(log n) c and diameter O(log 2 n) L. O(log n) c and diameter O(log 2 n) L. Remark: 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 O(log n,log n) Their ratio is O(log n,log n)

11. 11 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: Costs will be: Where Where

12. 12 So if you buy at bulk you save More generally, we have a non-decreasing monotone concave function g v : R  R for every v where g v (b) is the minimum cost of a router/switcher with bandwidth b. 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

13. 13 Ilustration of the cost-distance variant s2s2 t1t1 s1s1 1,6 2,9 t2t2 2,5 1,3 2,3 1,3 1,4 v u C(V')= c(s 1 )+c(u)+c(v)+c(w)+c(t 1 )+c(s 2 )+c(t 2 )=38 l(V')=2·l(u)+2 · l(v)+l(w)+l(s 1 )+l(t 1 )+l(s 2 )+l(t 2 )=13 w 2,8 2,4 2,3

14. 14 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 cost of the partial solution # of new pairs routed # of new pairs routed The algorithm tries to find low density partial solution at each iteration

15. 15 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 root r  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

16. 16 How do we improve?  Chandra et al proved: There is a junction tree with cost O(opt c /h) There is a junction tree with cost O(opt c /h) However the diameter is O(log n)· opt l /h However the diameter is O(log n)· opt l /h Note that there is an O(log n) advantage Note that there is an O(log n) advantage for the cost in this lemma. for the cost in this lemma.  There are what we call density LP that will induce a penalty of O(log 2 n) in the density of the actual tree found  As stated, O(log 3 n) density for length and one more O(log n) for set cover type payment, O(log 4 n) O(log n) for set cover type payment, O(log 4 n)

17. 17 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·opt l /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 log n factor than the opt set. BUT THIS HAS NO AFFECT AT LENGTH DENSITY

18. 18 Summary  Cost: Looses nothing from the junction tree lemma, looses O(log 2 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(log 3 n) times the cost and length optima

19. 19 Open Problems  Our guess is that MaxCT should have  (log n) lower bound (currently only  (log n) lower bound (currently only  (log log n) )  (log log n) )  Our guess is that MBB and Cost-Distance should have O(log 2 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? (O(log n),O(log n)) ratio. Proof anyone?