All-or-Nothing Multicommodity Flow

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All-or-Nothing Multicommodity Flow

• Chandra Chekuri Sanjeev Khanna Bruce
  Shepherd
• Bell Labs U. Penn Bell Labs

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Routing connections in networks
NY SF 10 Gb/sec NY SF 20 SE DE 5 SF DE
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SE
DE
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NY
Core Optical Network
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Multicommodity Routing Problem

• Network graph with edge capacities
• Requests k pairs, (si, ti) with demand di
• Objective find a feasible routing for all pairs
• Optimization maximize number of pairs routed

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All-or-Nothing Flow Problems

• Pair is routed only if all of di satisfied
• Single path for routing unsplittable flow
• (connection oriented networks)
• Fractional flow paths all-or-nothing flow
• (packet routing networks)
• Integer flow paths all-or-nothing integer flow
• (wavelength paths)

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Complexity of AN-Flow

• di 1 for all i
• Single path edge disjoint paths problem (EDP)
• classical problem, NP-hard
• only polynomial approx ratios
• AN-MCF APX-hard on trees
  approximation ?

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Approximating EDP/AN-MCF

• O(min(n2/3,m1/2)) approx in dir/undir graphs
  (EDP/UFP) Kleinberg 95, Srinivasan 97,
  Kolliopoulos-Stein 98, C-Khanna 03,
  Varadarajan-Venkataraman 04
• EDP is W(n1/2 - e)-hard to approx in directed
  graphs
• Guruswami-Khanna-Rajaraman-Shepherd-Yannakaki
  s 99
• LP integrality gap for EDP is W(n1/2) GVY 93
• AN-MCF APX-hard on trees
  Garg-Vazirani-Yannakakis 93

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Results

• In undirected graphs AN-MCF has an O(log3
  n log log n) approximation
• Polynomial factor to poly-logarithmic factor
• Approx via LP, integrality gap not large

For planar graphs O(log2 n log log n) approx Same
ratios for arbitrary demands dmax umin Online
algorithm with same ratio
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LP Relaxation

• xi amount of flow routed for pair (si, ti)
• max åi xi
• s.t
• xi flow is routed for (si,ti) 1 i k
• 0 xi 1 1 i k

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A Simple Fact

• Given AN-MCF instance all di 1
• Can find W(OPT) pairs such that each pair routes
  1/log n flow each
• How? rand rounding of LP and scaling down
• Problem we need pairs that send 1 unit each

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Nice Flow Paths

• Suppose all flow paths use a single vertex v

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Routing via Clustering

• cluster has log n terminals
• cluster induces a connected component
• clusters are edge disjoint

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Clustering

• Finding connected edge-disjoint clusters?
• G is connected use a spanning tree for a rough
  grouping of terminals
• New copy of G for clustering congestion 2
• 1 for clustering, 1 for routing
• Congestion 1 using complicated clustering

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How to find nice flow paths?

• Algorithmic tool
• Rackes hierarchical graph decomposition for
  oblivious routing Räcke02

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Räckes Graph Decomposition

• Represent G as a capacitated tree T

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10
4
v
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2
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leaves of T are vertices of G internal node v
G(v) is induced graph on leaves of T(v)
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Räckes Result

• T is a proxy for G
• For all D
• c(D,G) c(D,T) a(G) c(D,G)
• Routing in T is unique
• a(G) O(log3 n) Räcke 02
• a(G) O(log2 n log log n) Harrelson-Hildrum-Rao
  03

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Routing details
With each v there is distribution pv on G(v) s.t
åi 2 G(v) pv(i) 1 s distributes 1 unit of
flow to G(v) according to pv t distributes 1
unit of flow to G(v) according to pv

v
t
s
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Back to Nice Flow Paths
X(v) pairs with v as their least common ancestor
(lca)
G(v), pv
s1
t1
s2
s3
t2
t3
s4
t4
Routing in T
Routing in G
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Algorithm

• Find set of pairs X that can be routed in T (use
  tree algorithm GVY93,CMS03)
• Each pair (si,ti) in X has a level L(i)
• Choose level L at which most pairs turn
• Route pairs independently in subgraphs at L

v
L
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Algorithm contd

• v at L , X(v) pairs in X that turn at v
• Can route 1/a(G) flow for each pair in X(v) using
  nice flow paths
• Use clustering to route X(v)/a(G) pairs
• Approx ratio is
• a(G) depth(T) O(log3 n log log n)

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Open Problems

• Improve approximation ratio
• What is integrality gap of LP ?
  No super-constant gap known
• Extend ideas to EDP
• Recent result Poly-log approximation for EDP/UFP
  in planar graphs with congestion 3