1. Comparing Min-Cost and Min-Power Connectivity Problems
Guy Kortsarz
Rutgers University,
Camden, NJ

2. Motivation-Wireless Networks
Nodes in the network correspond to transmitters
More power  larger transmission range
transmitting to distance r requires r power, 2  r  4
Transmission range = disk centered at the node
Battery operated  power conservation critical
Type of problems:
Find min-power range assignment so that the resulting
communication network satisfies prescribed properties.

3. Directed Networks
Define costs c(e) that takes already into account the dependence on the distance . The cost c(e), e = (u,v) would be r with r the distance and the appropriate .
In general, power to send from u to v not the same as v to u
Thus power of v in directed graphs:
pE' (v)=Max{eE' leaves v}{c(e)}
For example: If no edge leaves v, p(v)=0
pE'( G)=∑v pE'(v)

4. Symmetric Networks pE' (v)=Max{eE' touching v}{c(e)}
Networks where the cost to send
from u to v or vise-versa is the same
Thus graph undirected and:
pE' (v)=Max{eE' touching v}{c(e)}
Many classical problems can and
have been studied with respect to the
(more difficult) min-power model

5. Communication networkb a c d g f e a b d g f e c
Range assignment
Communication network

6. EXAMPLE
UNIT COSTS
c(G) = n
p(G) = 1
c(G) = n
p(G) = n + 1

7. Example p(a) = 7, p(b) = 7, p(c) = 9, etc. b 7 a f 5 4 2 h 8 8 6 9 c d3 g

8. The Vertex k - Connectivity Problem
We are given an integer k
The goal is to make the graph resilient to at most k-1 station crashes
Design a min-power (min-cost)
subgraph G(V, E) so that every
u,v V admits at least k vertex-disjoint paths from u to v

9. Example
k=2 (2-connected graph) b a c

10. Previous Work for Min-Power Vertex k - Connectivity
Min-Power 2 Vertex-connectivity, heurisitic study [Ramanathan, Rosales-Hain, 2000]
11/3 approximation for k=2 (see easy 4 ratio later) [Kortsarz, Mirrokni, Nutov, Tsano, 2006]
Cone-Based Topology Control for Unit-Disk Graphs
[M. Bahramgiri, M. Hajiaghayi and V. Mirrokni, 2002]
O(k)-approximation Algorithm and a Distributed Algorithm for Geometric Graphs
[M. Hajiaghayi, N. Immorlica, V. Mirrokni, 2003]

11. Recent Result
Kortsarz, Mirrokni, Nutov, Tsano show that the vertex k-connectivity problem is ″almost″ equivalent with respect to approximation for cost and power (somewhat surprising)
In all other problem variants almost, the two problems behave quite differently
Based on a paper by
[M. Hajiaghayi, G. Kortsarz, V. Mirrokni and Z. Nutov, IPCO 2005]

12. Comparing Power And Cost Spanning Tree Case
The case k = 1 is the spanning tree case
Hence the min-cost version is the
minimum spanning tree problem
Min-power network: even this simple
case is NP-hard [Clementi, Penna, Silvestri, 2000]
Best known approximation ratio: 5/3
[E. Althaus, G. Calinescu, S.Prasad,
N. Tchervensky, A. Zelikovsky, 2004]

13. The case k=1: spanning tree
The minimum cost spanning tree is a ratio 2 approximation for min-power.
Due to: L. M. Kerousis, E. Kranakis, D. Krizank and A. Pelc, 2003

14. Spanning Tree (cont’) c(T)  p(T): Assign the parent edge ev to v
Clearly, p(v)  c(ev)
Taking the sum, the claim follows
p(G)  2c(G) (on any graph):
Assign to v its power edge ev
Every edge is assigned at most twice
The cost is at least
The power is at exactly

15. Relating the Min-Power and Min-Cost k - Connectivity Problems
An Edge e G is critical for k vertex-connectivity if G-e is not k vertex-connected
Theorem (Mader): In a cycle with every edge is critical there exists at least one vertex of degree k

16. Reduction to a Forest Solution
Say that we know how to approximate
by ratio  the following problem:
The Min-Power Edge-Multicover problem:
Input: G(V, E), c(e), degree
requirements r(v) for every v V
Required: A subgraph G(V, E) of
minimum power so that degG(v)  r(v)
Remark: polynomial problem for cost
version

17. Reduction to Forest (cont’)
Clearly, the power of a min-power Edge-Multicover solution for
r(v) = k-1 for every v is a lower bound on the optimum min-power
k-connected graph
Hence at cost at most opt we may start with minimum degree k -1

18. Reduction to Forest (cont’)
Let H be any feasible solution for the
Edge-Multicover problem with
r(v) = k-1 for all v
Claim: Let G = H + F with F any minimal augmentation of H into a k vertex-connected subgraph.
Then F is a forest

19. Reduction to Forest (cont’)
Proof:
Say that F has a cycle.
Consider a cycle C in F
All the edges of C are critical in H + F
By Mader’s theorem there must be a
vertex v in the cycle with degree k
But H(C) = k - 1, thus
(H+F)(C)  k+1, contradiction

20. Comparing the Cost and the Power
Theorem: If MCKK admits an  approximation then MPKK admits  + 2  approximation.
Similarly: approximation for min-power k-connectivity gives  +  approximation for min-cost
k - connectivity [M. Hajiaghayi, G. Kortsarz, V. Mirrokni and Z. Nutov, 2005]
Proof: Start with a β approximation H for the min-power vertex r(v) = k-1 cover problem
Apply the best min-cost approximation to turn H to a minimum cost vertex k - connected subgraph H + F, F minimal

21. Comparing the Cost and the Power (cont’)
Since F is minimal, by Mader’s theorem F is a forest
Let F* be the optimum augmentation. Then the following inequalities hold:
1) c(F)   c(F*) (this holds because  approximation)
2) p(F)  2c(F) (always true)
3) c(F*)  p(F*) (F* is a forest);
4) p(F)  2c(F)  2c(F*)  2 p(F*) QED

22. Best Results Known for Min-Cost Vertex k - Connectivity
Simple k-ratio approximation
[G. Kortsarz, Z Nutov, 2000]
Undirected graphs, k  (n/6)1/2, O(log n) approximation
[J. Cheriyan, A.Vetta and S.Vempala, 2002]
For any k (directed graphs as well):
O(n/(n - k))log2k
[G. Kortsarz and Z. Nutov, 2004]
For k = n - o(n), k1/2

23. Approximating the Min-Power Edge - Multicover Problem and Related Variants
Example: some versions may be difficult.
Say that we are given a budget k and all requirements are at least k - 1. All edge costs are 1.
Required: a subgraph of power at most k that meets the maximum requirement possible.

24. Approximating the Min-Power Edge- Multiover Problem (cont’)
The problem resulting is the densest
k-subgraph problem
Best known ratio:
n 1/3 - 
for  about 1/60
[U. Feige, G. Kortsarz and D. Peleg, 1996]

25. Approximating Edge-Multicover
Very hard technical difficulty: Any edge adds power to both sides.
Because of that: take k best edges, ratio k
Usefull first reduction: 3 a b
c’’
d’’
a’’
b’’ 6 6 6 8 8 5 3 8 3 d 5 c 5 d’ a’ b’ C’

26. An Overview Hence assume input B(X,Y,E) bipartite.
Only Y have demands.
However: both X and Y have costs
Assume opt is known
Main idea: Find F so that:
pF(V)  3opt
rF(B)  (1 - 1/e)  r(B) / 2
Clearly, this implies O(log n) ratio as
r(B)=O(n2)

27. Reduction to a Special Variant of the Max-Coverage Problem
Let R = r(Y)
The edge e = (x,y) is dangerous if
cost(e)  2opt r(y)/R;
A dangerous edge requires more than twice “its share” of the cost
Dangerous edges can be “ignored”; They cover at most half the demand.
Thus

28. The Cost Incurred by Non-Dangerous Edges
Since no dangerous edges used the cost is at most
Hence, focus on non-dangerous edges because even if every yY is touched by its heaviest (non-dangerous) edge the total cost on the Y side is O(opt).
Only try to minimize the cost invoked at X
This is reducible to a generalization of set-coverage

29. The Max-Coverage Problem With Group Budget Constrains
Select at most one of the following sets: 2 5 7 1 1 2 5 7 1 1 1 2 5 2
C=1
C=2
C=7
C=5

30. Approximating Set-Coverage with Group Budget Constrains
We reduced to a problem similar to the max-coverage algorithm
However, we have group constrains:
sets are split into groups. At most one set
can be selected of every group
Can be approximated within (1-1/e)
By pipage rounding [Ageev,Sviridenko 2000]
Invest opt, cover (1-1/e)/2 of the demand
O(log n) ratio approximation

31. General Requirements In the most general case:
requirement r (u, v) for every
u, v  V.
r (u, v) = 7 means 7 vertex disjoint
paths from u to v are required.

32. The Steiner Network Problem Vertex Version
Input: G ( V, E ), costs c(e) for every edge e E
requirements r(u,v) for every u,v  V
Required: A subgraph G′ ( V, E′ ) of G so that G′ has
r(u,v) vertex disjoint uv-paths for all u,v  V
Usual Goal: Mnimize the cost,
Alternative Goal: Minimize the power

33. Previous Work on Steiner Network
The edge + sum version admits 2 approximation.
[Jain, 1998].
The algorithm of Jain: Every BFS has an entry of value at least ½. Hence, iterative rounding.
The min-cost Steiner network problem vertex
version admits no
ratio approximation unless NP  DTIME(npolylog n) ,
[Kortsarz, Krauthgamer and Lee, 2002]
The result is based on 1R2P with projection property

34. Remarks Only Max-SNP hardness is known for min-power edge-coverage
For general rij only 4 rmax upper bound is known, [KMNT]
The edge case admits n1/2 approximation [HKMN]
Directed variants: even k edge-disjoint
path from x to y 1R2p Hard [KMNT]

35. Open Problems
The case r(u,v) {0, 1}. We recently broke the obvious ratio 4 (any solution is a forest so use ratio 2 for min-cost to get 22=4). Our ratio is 11/3. What is the best ratio?
Does min-cost (min-power) vertex k-connectivity admit (log n) lower bound?
This problem related to deep concepts in graphs known as critical graphs
Does the min-power edge-multicover problem admit an (log n) lower bound?
Can we give polylog for k vertex-connectivity directed graphs?