1. Algorithms for Routing Node-Disjoint Paths in Grids
Rauting
This is a joint work with Julia and david
Julia Chuzhoy David Kim Rachit Nimavat

2. Node-Disjoint Paths (NDP)
Input:
Graph G (n vertices)
Source-Target pairs 𝑠 1 , 𝑡 1 , 𝑠 2 , 𝑡 2 ,…, 𝑠 𝑘 , 𝑡 𝑘
Goal:
Route as many “demand pairs” as possible
Routing should be node-disjoint
OPT : 2

3. Node-Disjoint Paths (NDP)

4. Complexity of NDP 2 2 . . . . }𝑘 k=2: Fixed k: Arbitrary k:
NP-hard in directed graphs [Fortune, Hopcroft, Wyllie ‘80]
Easy in undirected graphs [Robertson-Seymour ‘90]
Fixed k:
Easy
Running time: f(k) * 𝑛 [Kawarbayashi, Kobayashi, Reed ‘12]
Arbitrary k:
NP-hard [Knuth, Karp ‘72] }𝑘
𝑓 5 ≈ 10 20,000
This dependence on k is bad

5. Complexity of NDP : Arbitrary k
NP-hard on undirected graphs [Knuth, Karp ‘72]
NP-hard on planar graphs [Lynch ’75]
NP-hard on grids [Kramer, Leeuwen ’84]
What Next?

6. Approximation Algorithms : Maximization Version
Efficient Algorithm
Gives a feasible solution of value OPT/𝛼
Smaller 𝛼 ⇒ Better algorithm
Hardness of Approximation
Approximation Algorithms 1 n
Best 𝛼
(e.g.: 𝑃≠𝑁𝑃)
Clearly alpha is lower bounded by 1. The question is what is the best alpha that you can achieve under standard complexity theoretic assumptions?
To give a lower bound on such alpha, there is hardness of approximation
Any approximation algorithm gives upper bound
One of the question in approx algo is to close the gap and find best alpha under standard complexity theory assumptions

7. Approximation Algorithm: Status
𝑂 𝑛 - approximation [Kolliopoulos, Stein ‘98]
Recent Improvements:
Planar Graphs : O 𝑛 9/19 - approximation [Chuzhoy, Kim, Li ‘16]
Grid Graphs: O ( 𝑛 1/4 )- approximation [Chuzhoy, Kim ‘15]
This Work:
*Grid Graphs: 2 𝑂 log 𝑛 - approximation
Check out the poster!
Recently Julia and david broke this 15 year \sqrt n barrier on grid graphs
And generalized the technique for planar graphs, with Shi Li: who was also here at ttic

8. Approximation Algorithm: Status
This Work:
Grid Graphs: 2 𝑂 log 𝑛 - approximation
(Restricted positioning of terminals)
Optimal?
Only 1.01-hardness known for grids
2 Ω log 𝑛 -hardness for “grids with holes” [Chuzhoy, Kim, N. ‘16]
log 100 𝑛< 2 𝑂 log 𝑛 < 𝑛 0.001

9. NDP on Grids: 3 Types of Demand Pairs
Standard multicommodity flow relaxation ⇒Ω( 𝑛 ) approximation 
S, T far from boundary
S, T close to boundary
S close to
boundary,
T far from it
There is a standard multicommodity flow relaxation for such routing problems in which we are allowed to send fractional flow on paths
Integrality gap for such reductions suggests that the approach of directly finding paths is not too practical.
Instead, write an LP to select a subset of demand pairs with some special property and route independently using that special property
From now on, we will focus on second case
One more reason to focus on second case is that the previous poly n approx algorithm gives a very good approximation for the first cases, but fails for the remaining two cases
Sources: S Targets: T

10. NDP on Grids: Type 2 s1 s2 s3 Sources on top boundary
Targets on a single row far from boundary
Goal: Routing on Node- Disjoint Paths
To make our lives simpler we assume that: t2 t3 t1

11. 1-distance Property
Space for escape?
𝑁 ′
> 𝑁 ′

12. 1-distance Property Space for escape?
Not really, the paths can be really convoluted
𝑁 ′
Thm: There is a subset of 𝑂𝑃𝑇 2 𝑂 log 𝑛 demand pairs with 1-distance property ∼0

13. log 𝑛 -distance Property
𝑁 ′
≫ 𝑁 ′ log 𝑛

14. log 𝑛 -distance Property
Gap of ≫𝑘 between Blue/Green terminals!
𝑘 2 (1− 1 log 𝑛 )
𝑘 log 𝑛
𝑘 2 (1− 1 log 𝑛 )
Sources:
2 disjoint small instances?
Destinations: ≫𝑘 ~𝑘

15. log 𝑛 -distance Property
k 2 (1− 1 log n )
Ans(1) = 1
Sources:
Ans(k) =
2*Ans( 𝑘 2 (1− 1 log 𝑛 ))
≈ 𝑘 log 𝑛
Destinations: ≫𝑘 ~𝑘

16. Thm: Can efficiently find routing of 𝑂𝑃𝑇 2 𝑂 log 𝑛 demand pairs
NDP on Grids: So Far
Step 1: There is a large subset with 1-distance property
Step 2: If only we can find such large subset…
Step 3: 1-distance property => Routing of a large subset
Thm: Can efficiently find routing of 𝑂𝑃𝑇 2 𝑂 log 𝑛 demand pairs

17. log 𝑛 -distance Property: Requirements
Large Hierarchical set of demand pairs,
such that:
Children are roughly of same size
Children are “well separated”

18. Simulating log 𝑛 -distance Property
𝑀 21
𝑑≫ 𝑀 1
𝑑≫ 𝑀
𝑑≫ 𝑀 2
𝑀 22
𝑀 2 𝑀
𝑀 1
𝑀 11
𝑀 12
Large Hierarchical set of demand pairs,
such that:
Children are roughly of same size
Children are “well separated”

19. Simulating log 𝑛 -distance Property With LP
𝑀 21
𝑑≫ 𝑀 1
𝑑≫ 𝑀
𝑑≫ 𝑀 2
𝑀 22
𝑀 2 𝑀
𝑀 1
𝑀 11
𝑀 12
Large Hierarchical set of demand pairs,
such that:
Children are roughly of same size
Children are “well separated”
Can write a Linear Program!
(And round it)

20. Future Work Thank You! Dealing with the Third Case
(When Sources and Targets, both are far from boundary)
Closing gap for grids
Only APX-hardness is known
Thank You!

21. Approximation Algorithm: 𝛼= 𝑛 [Kolliopoulos, Stein ‘98]
While there is a NDP connecting any demand pair:
Add such shortest path P
Delete from OPT all paths sharing vertices with P
What if length of P > 𝑛 ?
𝑂𝑃𝑇< 𝑛
What if length of P < 𝑛 ?
At most 𝑛 paths in OPT deleted
Continue!
𝛼= 𝑛