Comparing Min-Cost and Min-Power Connectivity Problems

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Comparing Min-Cost and Min-Power Connectivity
Problems

• Guy Kortsarz
• Rutgers University,
• Camden, NJ

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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.

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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)Maxe?E' leaves vc(e)
• For example If no edge leaves v, p(v)0
  
• pE'( G)?v pE'(v)

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Symmetric Networks

• Networks where the cost to send
• from u to v or vise-versa is the same
• Thus graph undirected and
• pE' (v)Maxe?E' touching vc(e)
• Many classical problems can and
• have been studied with respect to the
• (more difficult) min-power model

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Range assignment
Communication network
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EXAMPLE UNIT COSTS
c(G) n p(G) 1
c(G) n p(G) n 1
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Why Not Complete Network?
b
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4
c
a
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a is not directly connected to c. Total power is
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Example

• p(a) 7, p(b) 7, p(c) 9, etc.

b
7
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Requirements

• Resilience to node-failures
• (node- connectivity problems)
• 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
• Edge-disjointness not very relevant

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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

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Example

• r(u, v) 2

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a
c
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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

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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

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Previous Work for Min-Power Vertex k -
Connectivity

• Min-Power 2 Vertex-connectivity, heurisitic study
  Ramanathan, Rosales-Hain, 2000
• 11/3 approximation for k2 (see easy 4 ratio
  later) Kortsarz, Mirrokni, Nutov, Tsano, 2006
• O(k) approximation
• M. Bahramgiri, M. Hajiaghayi and V.
  Mirrokni, 2002,
• M. Hajiaghayi, N. Immorlica, V. Mirrokni
  2003

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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

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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

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The case k1 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

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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

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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

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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

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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

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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

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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 Maders theorem there must be a
• vertex v in the cycle with degree k
• But ?H(C) k - 1, thus
• ?(HF)(C) ? k1, contradiction

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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 ß 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

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Comparing the Cost and the Power (cont)

• Since F is minimal, by Maders 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

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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

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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.

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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

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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

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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?) ? 3?opt
• rF(B) ? (1 - 1/e) ? r(B) / 2
• Clearly, this implies O(log n) ratio as
• r(B)O(n2)

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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

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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

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The Max-CoverageProblem With Group Budget
Constrains

• Select at most one of the following sets

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7
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C2
C7
C1
C5
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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

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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

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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?24). 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?