multicommodity flow, well-linked terminals and routing problems chandra chekuri lucent bell labs...
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Multicommodity flow, well-linked terminals and routing problems
Chandra ChekuriLucent 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 disjointNDP: paths are node disjointAN-Flow: flow of one unit per pair with
edge/node capacity equal to 1
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Disjoint paths vs An-Flow
s1 s2
t1t2
s1
s2
t1t2
1/2
1/2
1/2
1/2
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Setup
Terminals: X = {s1,t1,s2,t2,...,sk,tk}each terminal occurs in exactly one pair, |X|
= 2kPairs: matching M on XInstance: (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 otherwisexi : 1 if siti is routed, 0 otherwise
max i xi s.t
xi = p 2 P(i) f(p) 1 · i · k
p: 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 Pxi : amount of flow routed for siti
max i xi s.t
xi = p 2 P(i) f(p) 1 · i · k
p: 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 : V ! R+
implicitly defines d with dij = (i)(j) / (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 = |(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 ¸ 1d is feasible in G if sparsity = (log n)[LR88,LLR94,AR94]
For product MC Flow in planar G, sparisty of (1) sufficient [KPR93]
Flow-cut gap (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
S V - S
for all S ½ V with |S Å X| · |X|/2, |(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 (u) = 1 if u 2 X
= 0 otherwise
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Cut vs Flow well-linked
X flow-linked ) X is ~cut-linkedX cut-linked ) X flow-linked with congestion
(G)
(G) – worst case flow-cut gap for product multicommodity instances in G
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Weighted versions
: X ! R+ weight function on X(v) : weight of v in X
-cut-linked: for all S ½ V with (S Å X) · (X)/2, |(S)| ¸ (S Å X)
-flow-linked: multicommodity flow instance with d(uv) = (u) (v) / (X) is feasible in G
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Well-linked instance of EDP
Input instance: G, X, MX = {s1, t1, s2, t2, ..., sk, tk} – terminal setM : 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, MX = {s1, t1, s2, t2, ..., sk, tk} – terminal setM : matching on X
(s1,t1), (s2,t2) ... (sk,tk)
X is -well-linked in G for some : X ! R+
Assume: (v) · 1
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Examples
s1 t1
s2 t2
s3 t3
s4 t4
Not a well-linked instance
s1 t1
s2 t2
s3 t3
s4 t4
A well-linked instance
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Well-linked Decomposition
G, X, M
G1, X1, M1
Mi ½ M
Xi is well-linked in Gi
i |Xi| ¸ OPT/
G2, X2, M2
Gr, Xr, Mr
edge disjoint subgraphs
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Example
s1 t1
s2 t2
s3 t3
s4 t4
s1 t1
s2 t2
s3 t3
s4 t4
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Well-linked Decomposition via Flow
G, X, M
Flow f
G1, X1, M1
Xi is i-flow-well-linked in Gi
i i(Xi) ¸ f/
G2, X2, M2
Gr, Xr, Mr
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Decomposition via trees/Racke
Simple decomposition for trees: = O(1)Represent G as a tree (approximately)
[Racke03]Done in [CKS04]
Decomposition based on recursive cuts [CKS05]simplebetter ratioapplies to node problems
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Trees
Define : X ! R+
(sj) = (tj) = fj the flow in LP
Suppose X is /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
S
V - S
ce < (S)/10
terminals in S are -well-linked!
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Decomposition using Sparse Cuts
Start with LP soln for given instancefj flow for pair sjtj : assume flow
decompositionf = j fj total flow in LP
define : X ! R+
(sj) = (tj) = fj
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Decomposition Algorithm
If X is / 10 (G) log k-flow-linked STOP
ElseFind a (approx) sparse cut (S,V-S) wrt in GRemove flow on edges of G(S)
G1 = G[S], G2 = G[V-S]
Recurse on G1, G2 with remaining flow
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Analysis
Remaining graphs at end of recursion(G1,X1,1) , (G2,X2,2) , ...., (Gh, Xh, h)
i is the remaining flow for Xi
Xi is i /10 (G) log k flow-linked in G_i
i i(Xi) ¸ Original flow - # edges cut
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Bounding the number of edges cut
X is not / 10 (G) log k flow-linked
) |G(S)| · (S) / 10 log k
S V - S
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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
i i (Xi) ¸ f/2
Xi is i/10 (G) log k flow-well-linked
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Fractional to integer well-linked
Theorem: G, X, M input instance. X is -flow-well-
linked. Then G, X’, M’ s.t
M’ ½ M, X’ is flow-well-linked |X’| = ((X))
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Edge case: spanning tree clustering
T spanning tree of G, rooted at rTv : subtree rooted at v
Can assume maximum degree of T is 4
1. Find deepest node u s.t (Tu) ¸ 1
Note:(Tu) · 5
2. Remove Tu from T3. Continue until (T) · 1
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Spanning tree clustering
0.5 0.7 0.1 0.3
0.4
0.2
0.3
0.8 0.4
0.4
0.6
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Spanning tree clustering
0.5 0.7 0.1 0.3
0.4
0.2
0.3
0.8 0.4
0.4
0.6
0.5 0.7
0.2
0.3
0.8 0.40.1 0.3
0.4
0.4
0.6
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Tree clustering
T1, T2, ..., Th clusters
Claim: h = ((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
0.5 0.7
0.2
0.3
0.1 0.3
0.4
0.4
0.6
0.8 0.4
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Representatives
Need representatives Y such that Y ½ Xi
Y induces a large submatching of Mi
Simple greedy scheme workspick 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 caseUse node-separators instead of edge separators
Clustering is not straighforward (can’t 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 (log1/2 n) factor
Implicitly from integrality gap results for all-or-nothing flow problem [Chuzhoy-Khanna05]
Conjecture: (log n) factor lower bound
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Flows, Cuts, and Integer Flows
max integer flow
max frac flow min multicut · ·
NP-hard NP-hardSolvable
Flow-cut gap thms [LR88 ...]??
+ 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 (log n) ?
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Open problems
Improve upper/lower bounds on well-linked decomposition. (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.tT(G) approximates G w.r.t sparse cutsApproximation factor – O((G) log n log log n)
[Harrelson-Hildrum-Rao04]
Apply algo. on T(G) to get decomposition for GLoss: polylog(n)