Comparing Min-Cost and Min-Power Connectivity

Problems

Guy KortsarzRutgers University,

Camden, NJ

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.

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 uThus power of v in directed graphs:

pE' (v)=Max{eE' leaves v}{c(e)} For example: If no edge leaves v, p(v)=0

pE'( G)=∑v pE'(v)

Symmetric Networks

Networks where the cost to send

from u to v or vise-versa is the same

Thus graph undirected and:

pE' (v)=Max{eE' touching v}{c(e)}

Many classical problems can and

have been studied with respect to the

(more difficult) min-power model

b

a

c

d

g

f

e

a

b

d

g

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e

c

Range assignment Communication network

c(G) = n

p(G) = n + 1

c(G) = n

p(G) = 1

EXAMPLE

UNIT COSTS

Example

p(a) = 7, p(b) = 7, p(c) = 9, etc.

7

5 8

9

8

54 2

3

6

a

b

c d

f

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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

Example

k=2 (2-connected graph)

a

b

c

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]

Recent ResultKortsarz, 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]

Comparing Power And Cost Spanning Tree Case

The case k = 1 is the spanning tree caseHence the min-cost version is the

minimum spanning tree problemMin-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]

The case k=1: spanning treeThe 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

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

v

vec

2

)(

v

vec )(

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

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

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

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

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

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

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) 2c(F*) 2 p(F*) QED

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

[G. Kortsarz and Z. Nutov, 2004]

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.

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]

Very hard technical difficulty: Any edge adds power to both sides.

Because of that: take k best edges, ratio k

Usefull first reduction:

a b

c d

3

6 8

a’ b’ C’ d’

a’’ b’’ c’’ d’’

3

3

6 6

8

Approximating Edge-Multicover

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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) 3opt

rF(B) (1 - 1/e) r(B) / 2

Clearly, this implies O(log n) ratio as

r(B)=O(n2)

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

optR

yropt

Yy

)(2

2)(

Ryr

Yy

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 yY 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

optR

yropt

Yy

2)(2

The Max-CoverageProblem With Group Budget

ConstrainsSelect at most one of the following sets:

1 2 7

1 1 2 5 1 7

C=5C=2 C=7C=1

2 512

5

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

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.

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

Ee

ecEc )()(

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

n1log2

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]

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 22=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?