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:
Rutgers University, Camden, NJ
Joint work with
The Open University, Israel
Multicommodity Buy At Bulk with Node
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
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:
Call this problem MBB
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’
c(V’)+ s,t dst distV’(s,t)
v induce a cost of l(v)·d. This is called the INCREMENTAL COST.
MBB and Cots-Distance was O(log4 n).
Given a graph G with vertex costs vertex profits and budget bound B, find a maximum profit subtree T Gof budget at most B
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
for any universal constant c even if the solution is allowed to violate the budget within any universal constant
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
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)
Costs will be:
Question: Given a set of bandwidth demands between nodes, install sufficient capacities at minimum cost. The cost per v is non-decreasing concave
cost of the partial solution
# of new pairs routed
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.
O(log n) for set cover type payment, O(log4n)
log n factor than the opt set. BUT THIS HAS NO AFFECT AT LENGTH DENSITY
(log n) lower bound (currently only
(log log n) )
(O(log n),O(log n)) ratio. Proof anyone?