1. Multipath Routing for Congestion Minimization & Multiroute Flows
Chandra Chekuri
Univ of Illinois, Urbana-Champaign
Joint work with Mark Idleman

2. s-t flows in directed graphs
G=(V,E) directed graph with edge capacities c(e), e in E
Edge based definition of flow
f(e) flow on edge e
flow conserved at internal nodes V \ {s,t}
f(e) ≤ c(e) for all e
Path based definition of flow
f(p) flow on path p ∈ 𝒫(st)
𝑝∋𝑒 𝑓 𝑝 ≤𝑐 𝑒 for all e

3. Flow Decomposition
Edge flow f of value v can be decomposed into a path flow in O(nm) time
A path flow of value v induces an edge flow of same value

4. Multi-route flows
𝒫(st) = { p | p is a st path } 𝒫(st, h) = {p = (p1,p2,...,ph) | each pj ∈ 𝒫(st) and the paths are edge-disjoint } h-route s-t flow f : 𝒫(st, h) → R+ f(p) flow on path-tuple p

5. s p q t

6. Multiroute flows: basic theorem
[Kishimoto,Aggarwal-Orlin] Theorem: An acyclic edge s-t flow x : E → R+ with value v can be decomposed into a h-route flow iff x(e) ≤ v/h for all edges e 3 2 1 1 s t s t 1 1

7. Congestion Minimization
Choose a path for each pair
Minimize max number of paths using any edge (congestion) G

8. Congestion Minimization
0.7
Choose a path for each pair
Minimize max number of paths using any edge (congestion)
Special case: Edge-Disjoint Paths
[Raghavan-Thompson’87]
Solve mc-flow relaxation (LP)
Randomly pick a path according to fractional solution
Chernoff bounds to show approx ratio of O(log n/log log n)
0.5
0.3
0.25 G
0.1
0.15
0.2
0.15
0.65

9. Multipath Routing h2 = 1 s3 s2 s1 t1 t2 t3
Choose hi paths for pair (si, ti)
(assume paths for pair disjoint)
Minimize max number of paths using any edge (congestion) G
h1 = 2
h3 = 2

10. Multipath Routing s3 s2 s1 t1 t2 t3 0.7
Choose hi paths for pair (si, ti)
(assume paths for pair disjoint)
Minimize max number of paths using any edge (congestion)
[Srinivasan’99]
Solve relaxation (LP)
Dependent rounding
O(log n/log log n) approx via negative correlation
0.5
0.3
0.3 G
0.25
0.95
0.7
0.5
0.8

11. Can use multiroute flow decomposition
Decompose for each pair flow into hi-route flow of value 1
Do independent rounding ala Raghavan-Thompson
0.5
0.3 s1 t1
0.25
0.95
0.2
0.25
0.05
0.5

12. Advantages
Rounding and analysis exactly the same as it is for standard congestion minimization. No need for technicalities of dependent rounding.
If paths are short can use Lovasz-Local-Lemma to reduce congestion to O(log d/log log d) where d is max # of edges in any path collection of decomposition (via existing results of [Srinivasan, Moser-Tardos, Haeupler-Saha-Srinivasan])

13. Choosing Paths
G=(V,E) , (s1,t1), …, (sk,tk) and need hi edge-disjoint paths per pair
Minimize congestion

14. Multi-route flow LP Min C  p ∈ P(siti, h) f(p) ≥ 1 for all (si, ti)
 p ∈ P(siti, h) :e ∈ p f(p) ≤ C for all e, i
f(p) ≥ 0 for all p

15. Multi-route flow LP Min C  p ∈ P(siti, hi) f(p) ≥ 1 for all (si, ti)
 p ∈ P(siti, hi) :e ∈ p f(p) ≤ C for all e, i
f(p) ≥ 0 for all p
How do we solve LP?
Ellipsoid for dual: separation oracle is min-cost flow
Compact LP followed by multi-route flow decomposition

16. Compact LP
Min C
for each i flow of value hi from si to ti
f(e,i) ≤ 1 for all e, i
∑i f(e,i) ≤ C for all e
f(e,i) ≥ 0 for all e, i
Can decompose f(e,i) into multi-route hi-route-flow and choose one “path collection” for each (si,ti) independently

17. Compact LP Min C for each i flow of value hi from si to ti
f(e,i) ≤ 1 for all e, i
∑i f(e,i) ≤ C for all e
f(e,i) ≥ 0 for all e, i
[Doerr-Kunnemann-Wahlstrom’10] solve same LP but do “standard” flow decomposition which does not guarantee disjoint paths. They employ dependent rounding after standard flow decomposition.

18. Choosing Short Paths
G=(V,E) , (s1,t1), …, (sk,tk) and need hi edge-disjoint paths per pair
Want paths for each pair to be “short”: no more than d edges in total
Minimize congestion

19. Decompose f(e,i) into multi-route hi-route-flow.
Min C
for each i flow of value hi from si to ti
f(e,i) ≤ 1 for all e, i
∑i f(e,i) ≤ C for all e
f(e,i) ≥ 0 for all e, I
∑e f(e,i) ≤ d for all i
Decompose f(e,i) into multi-route hi-route-flow.
Apply filtering to find flow of value at least ½ on paths which use at most 2d edges
Choose one “path collection” for each (si,ti) independently from short path collections
Congestion of O(log d/log log d) via LLL

20. Concluding Remarks Multiroute flows are useful to know
Several explicit and implicit applications in routing and network design – see refs in paper

21. Thank You!