Multicommodity flow, well-linked terminals and routing problems

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Multicommodity flow, well-linked terminals and
routing problems

• Chandra Chekuri
• Lucent Bell Labs

Joint work with Sanjeev Khanna and Bruce
Shepherd Mostly based on paper in STOC 05
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Routing Problems

• Input Graph G(V,E), node pairs s1t1, s2t2, ...,
  sktk
• Goal Route a maximum of si-ti pairs
• Route?
• EDP path for each pair, paths edge disjoint
• NDP paths are node disjoint
• AN-Flow flow of one unit per pair with edge/node
  capacity equal to 1

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Disjoint paths vs An-Flow
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Setup

• Terminals X s1,t1,s2,t2,...,sk,tk
• each terminal occurs in exactly one pair, X
  2k
• Pairs matching M on X
• Instance (G,X,M)
• unit capacity graph
• Focus edge problems, EDP and An-flow.

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Multicommodity Flow Formulation (IP)

• P(i) set of paths between si and ti
• P P(1) P(2) ... P(k)
• f(p) 1 if flow on path p 2 P, 0 otherwise
• xi 1 if siti is routed, 0 otherwise
• max åi xi s.t
• xi åp 2 P(i) f(p) 1 i k
• åp e 2 p f(p) 1 e 2 E
• xi, f(p) 2 0,1

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Multicommodity Flow Formulation (LP)

• P(i) set of paths between si and ti
• P P(1) P(2) ... P(k)
• f(p) flow on path p 2 P
• xi amount of flow routed for siti
• max åi xi s.t
• xi åp 2 P(i) f(p) 1 i k
• åp e 2 p f(p) 1 e 2 E
• xi, f(p) 2 0,1

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Framework

1. Start with an LP solution.
2. Use LP solution to decompose the input instance
   into a collection well-linked instances.
3. Use well-linkedness to route large fraction

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Outline

• Cut vs Flow well-linkedness
• Well-linked decomposition
• Multicommodity flow to well-linked decomp
• decomposition via cuts
• fractional well-linkedness to well-linkedness
• Conclusions

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Multicommodity Flows

• MC Flow instance
• capacitated graph G
• non-negative demand matrix d on V x V
• route dij flow for node pair ij
• Product MC Flow instance
• node weights p V ! R
• implicitly defines d with dij p(i)p(j) / p(V)

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Sparse Cuts and Multicomm. Flow

• Given a cut (S, V-S) in G and demand matrix d
• sparsity of S d(S) / d(S,V-S)
• MCflow for d is feasible in G implies sparsity 1

S
V - S
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Sparse Cuts and MC Flow

• MCflow for d is feasible in G implies sparsity
  1
• d is feasible in G if sparsity W(log n)
• LR88,LLR94,AR94
• For product MC Flow in planar G, sparisty of W(1)
  sufficient KPR93
• Flow-cut gap b(G) minimum sparsity reqd for
  guaranteeing mc flow

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Cut-Well-linked Set

• Subset X is cut-well-linked in G if for every
  partition (S,V-S) , of edges cut is at least
  of X vertices in smaller side

for all S ½ V with S Å X X/2, d(S) S
Å X
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Flow-Well-linked Set

• Subset X is flow-well-linked in G if the
  following multicommodity flow is feasible in G
• for u,v in X, d(uv) 1/X
• product (uniform) multicommodity flow on X
• p(u) 1 if u 2 X
• 0 otherwise

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Cut vs Flow well-linked

• X flow-linked ) X is cut-linked
• X cut-linked ) X flow-linked with congestion b(G)
• b(G) worst case flow-cut gap for product
  multicommodity instances in G

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Weighted versions

• p X ! R weight function on X
• p(v) weight of v in X
• p-cut-linked for all S ½ V with p(S Å X)
  p(X)/2, d(S) p(S Å X)
• p-flow-linked multicommodity flow instance with
  d(uv) p(u) p(v) / p(X) is feasible in G

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Well-linked instance of EDP

• Input instance G, X, M
• X s1, t1, s2, t2, ..., sk, tk terminal set
• M matching on X
• (s1,t1), (s2,t2) ... (sk,tk)
• X is well-linked in G

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Well-linked instance weighted

• Input instance G, X, M
• X s1, t1, s2, t2, ..., sk, tk terminal set
• M matching on X
• (s1,t1), (s2,t2) ... (sk,tk)
• X is p-well-linked in G for some p X ! R
• Assume p(v) 1

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Examples
Not a well-linked instance
A well-linked instance
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Well-linked Decomposition
G, X, M
edge disjoint subgraphs
G1, X1, M1
G2, X2, M2
Gr, Xr, Mr
Mi ½ M Xi is well-linked in Gi
åi Xi OPT/a
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Example
s2
t2
s2
t2
s3
t3
s4
t4
s3
t3
s4
t4
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Well-linked Decomposition via Flow
G, X, M Flow f
G1, X1, M1
G2, X2, M2
Gr, Xr, Mr
Xi is pi-flow-well-linked in Gi
åi pi(Xi) f/a
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Decomposition via trees/Racke

• Simple decomposition for trees a O(1)
• Represent G as a tree (approximately) Racke03
• Done in CKS04
• Decomposition based on recursive cuts CKS05
• simple
• better ratio
• applies to node problems

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Trees

• Define p X ! R
• p(sj) p(tj) fj the flow in LP
• Suppose X is p/10-flow-well-linked done!
• Otherwise exists cut of sparsity less than 1/10
• Pick sparse cut (S,V-S) with S minimal

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Trees
ce lt p(S)/10
V - S
S
terminals in S are p-well-linked!
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Decomposition using Sparse Cuts

• Start with LP soln for given instance
• fj flow for pair sjtj assume flow decomposition
• f åj fj total flow in LP
• define p X ! R
• p(sj) p(tj) fj

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Decomposition Algorithm

• If X is p / 10 b(G) log k-flow-linked STOP
• Else
• Find a (approx) sparse cut (S,V-S) wrt p in G
• Remove flow on edges of dG(S)
• G1 GS, G2 GV-S
• Recurse on G1, G2 with remaining flow

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Analysis

• Remaining graphs at end of recursion
• (G1,X1,p1) , (G2,X2,p2) , ...., (Gh, Xh, ph)
• pi is the remaining flow for Xi
• Xi is pi /10 b(G) log k flow-linked in G_i
• åi pi(Xi) Original flow - edges cut

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Bounding the number of edges cut

• X is not p / 10 b(G) log k flow-linked
• ) dG(S) p(S) / 10 log k

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Analysis cont

• Theorem total number of edge cut is f/2
• T(x) max of edges cut if started with flow x
• T(f) T(f1) T(f2) f1 / 10 log k
• For f k, T(f) f/2

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Analysis contd

• åi pi (Xi) f/2
• Xi is pi/10 b(G) log k flow-well-linked

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Fractional to integer well-linked

• Theorem
• G, X, M input instance. X is p-flow-well-linked.
• Then G, X, M s.t
• M ½ M,
• X is flow-well-linked
• X W(p(X))

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Edge case spanning tree clustering

• T spanning tree of G, rooted at r
• Tv subtree rooted at v
• Can assume maximum degree of T is 4
• 1. Find deepest node u s.t p(Tu) 1
• Note p(Tu) 5
• 2. Remove Tu from T
• 3. Continue until p(T) 1

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Spanning tree clustering
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Spanning tree clustering
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Tree clustering

• T1, T2, ..., Th clusters
• Claim h W(p(X))
• Y is a set of representatives if Y Å Ti 1 for
  all i
• Lemma Y is ½ - flow-well-linked

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Representatives are well-linked
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Representatives

• Need representatives Y such that
• Y ½ Xi
• Y induces a large submatching of Mi
• Simple greedy scheme works
• pick si and ti
• remove all terminals in trees of si and ti
• continue

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Node case

• Well-linked decomposition same as for edge case
• Use node-separators instead of edge separators
• Clustering is not straighforward (cant assume
  degree bound)
• In CKS05 weaker bounds than for edge case
• Recent work same as for edge case. More
  technically involved

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Lower Bounds

• Well-linked decomposition has to lose W(log1/2 n)
  factor
• Implicitly from integrality gap results for
  all-or-nothing flow problem Chuzhoy-Khanna05
• Conjecture W(log n) factor lower bound

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Flows, Cuts, and Integer Flows
NP-hard
NP-hard
Solvable
max integer flow
max frac flow
min multicut


graph theory
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Weaker decomp for planar graphs

• Well-linked decomp yields O(log n) approx for
  planar graph EDP (congestion 2)
• Recent result for planar EDP O(1) approx with
  congestion 4 CKS 05
• Weaker decomp based on planar graph properties.
• Q well-linked in planar loses W(log n) ?

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Open problems

• Improve upper/lower bounds on well-linked
  decomposition. Q(log n)?
• Approx algorithms for EDP/NDP in general graphs
  with congestion O(1)
• essentially reduced to a graph theory problem
• Directed graphs?

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Thank You!
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Trees to Graphs using Racke

• Hierarchical graph decomposition Racke03
• Given graph G, exists capacitated tree T(G) s.t
• T(G) approximates G w.r.t sparse cuts
• Approximation factor O(b(G) log n log log n)
  Harrelson-Hildrum-Rao04
• Apply algo. on T(G) to get decomposition for G
• Loss polylog(n)