New Algorithms for Disjoint
Paths Problems
Sanjeev Khanna
University of Pennsylvania
Joint work with
Chandra Chekuri
Bruce Shepherd
Edge Disjoint Path Problem (EDP)
Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk
Goal: Route a maximum # of si-ti pairs using
edge-disjoint paths
s3
s4
t3
s2
s1
t2
t4
t1
Edge Disjoint Path Problem (EDP)
Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk
Goal: Route a maximum # of si-ti pairs using
edge-disjoint paths
s3
s4
t3
s2
s1
t2
t4
t1
EDP on Stars
u1
u4
1
u1
u2
1
1
u2
1
u3
u3
Matching in G
And vice versa.
EDP in Star
u4
Two Pair Problem
Input: Graph G(V,E) and two pairs s1t1, s2t2.
Goal: Can we simultaneously route s1 to t1 and s2 to t2
in an edge-disjoint manner?
Two Pair Problem
Input: Graph G(V,E) and two pairs s1t1, s2t2.
Goal: Can we simultaneously route s1 to t1 and s2 to t2
in an edge-disjoint manner?
 NP-hard if G is a directed graph [Fortune, Hopcroft,
Wylie’80].
 Polynomial-time solvable for any constant number of
pairs if G is undirected [Roberston, Seymour’88].
Routing Problems




Related problems:
 node disjoint paths
 each pair si-ti has a demand di and edges/nodes
have capacities
Fundamental to combinatorial optimization
Applications to VLSI, network design and routing,
resource allocation & related areas
Related to significant theoretical advances
Coping with Hardness
Settle for sub-optimal solutions:
 route only a fraction of the pairs that can be
routed in an optimal solution
 allow for small violations of edge capacities
Approximation algorithm A
 runs in polynomial time
 approximation ratio: how good is A
 approx ratio b ) A(I) ¸ OPT(I)/b for all I
 Would like b to be as small as possible
A Greedy Algorithm
 Among the unrouted pairs, pick the pair that has the
shortest path in the current graph.
 Route this pair and remove all edges on the path from
the graph.
Clearly gives an edge-disjoint routing.
How good is this algorithm?
Analysis of the Greedy Algorithm
n: # of vertices
m: # of edges
Fix an optimal solution, say, OPT.
If each greedy path is at most m1/2 edges long, it
destroys at most m1/2 paths in OPT.
Suppose at some point, a path chosen by greedy
is longer than m1/2. Since there are only m edges,
OPT can chose at most m1/2 paths from here on.
So greedy gives an O(m1/2)-approximation.
A Bad Example
Greedy chooses the red path and none of the blue pairs
can be routed as a result.
Surely, we could do better ...
Not if the graph is directed!
[Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99]
It is NP-hard to get an O(m1/2 - e) approximation for
directed graphs for any e > 0.
Surely, we could do better ...
Not if the graph is directed!
[Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99]
It is NP-hard to get an O(m1/2 - e) approximation for
directed graphs for any e > 0.
[Chuzhoy, K ’05]
For undirected graphs, O( log1/2-e n)-approximation is hard.
(Builds on [Andrews, Zhang ’05] .)
Surely, we could do better ...
Not if the graph is directed!
[Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99]
It is NP-hard to get an O(m1/2 - e) approximation for
directed graphs for any e > 0.
[Chuzhoy, K ’05]
For undirected graphs, O( log1/2-e n)-approximation is hard.
(Builds on [Andrews, Zhang ’05] .)
The O(m1/2)-approximation is the best known in general
(as a function of m).
All-or-Nothing Flow Prob (ANF)
Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk.
Goal: Route a maximum # of si-ti pairs such that
each routed pair has one unit of flow.
s1
s2
s1
1/2
s2
1/2
t2
t1
t2
1/2
1/2
t1
Recent Progress
[Chekuri, K, Shepherd: ’04 and ’05]

O(log2 n) approximation for ANF in undirected graphs.
 [Chuzhoy, K ’05] O( log1/2-e n)-approximation is hard.
Recent Progress
[Chekuri, K, Shepherd: ’04 and ’05]

O(log2 n) approximation for ANF in undirected graphs.
 [Chuzhoy, K ’05] O( log1/2-e n)-approximation is hard.

O(log n) approximation for EDP in planar undirected
graphs when up to two paths can share an edge.
Recent Progress
[Chekuri, K, Shepherd: ’04 and ’05]

O(log2 n) approximation for ANF in undirected graphs.
 [Chuzhoy, K ’05] O( log1/2-e n)-approximation is hard.


O(log n) approximation for EDP in planar undirected
graphs when up to two paths can share an edge.
Similar results for the node-disjoint versions as well as
versions with arbitrary demands and capacities.
Recent Progress
[Chekuri, K, Shepherd: ’04 and ’05]

O(log2 n) approximation for ANF in undirected graphs.
 [Chuzhoy, K ’05] O( log1/2-e n)-approximation is hard.


O(log n) approximation for EDP in planar undirected
graphs when up to two paths can share an edge.
Similar results for the node-disjoint versions as well as
versions with arbitrary demands and capacities.
Previous algorithms had W(n1/2) approximation ratio.
Rest of the Talk
 EDP in planar graphs
 A fractional relaxation
 A new framework for routing problems
 Well-linked sets and crossbars
 Routing using crossbar structures
 EDP with congestion in general graphs
Multicommodity Flow Formulation (LP)
 Routing is relaxed to be a flow from si to ti.
 A pair can be routed for a fractional amount.
xi : fraction of si-ti flow that is routed.
Max i xi
s.t.
8e
total flow through e · 1.
0 · xi · 1.
Randomized Rounding
[Raghavan and Thompson ’87]
 For each pair (si,ti), decompose the flow xi 2 [0,1] into
a collection of flow paths.
 Decide to route pair (si,ti) with probability xi.
 If yes, choose an si-ti flow path with probability
proportional to the flow on it.
O(n1/c)-approximation if we allow up to c paths to use
an edge.
How Good is this LP?
[GVY ’93]
tk
tk-1
ti
t3
t2
t1
sk sk-1
si
s3 s2 s1
How Good is this LP?
[GVY ’93]
tk
tk-1
ti
W(n1/2) Lower Bound
t3
t2
t1
sk sk-1
si
s3 s2 s1
Gap holds for planar graphs
How Good is this LP?
[GVY ’93]
tk
tk-1
ti
W(n1/2) Lower Bound
t3
t2
O(n2/3) Upper Bound
t1
sk sk-1
si
s3 s2 s1
Gap holds for planar graphs
A New Framework
1. Solve the LP relaxation.
2. Use LP solution to decompose input instance
into a collection of instances with special
structure, called well-linked instances.
3. Well-linked instances have special properties;
use them for routing!
High-level Algorithm for Planar Graphs
1. Solve the LP relaxation.
2. Use LP solution to decompose input instance
into a collection of well-linked instances.
3. Well-linked planar instances have crossbars, use
them for routing!
Assume w.l.o.g. input graph to be bounded degree.
Well-linked Set
Subset X is 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, |d(S)| ¸ |S Å X|
Instance of EDP
Input instance: G, X, M
G : underlying graph.
X : {s1, t1, s2, t2, ..., sk, tk} is the terminal set
M : a matching on X , namely, (s1,t1), (s2,t2) ... (sk,tk)
that needs to be routed in G.
Well-linked Instance of EDP
Input instance: G, X, M
G : underlying graph.
X : {s1, t1, s2, t2, ..., sk, tk} is the terminal set
M : a matching on X , namely, (s1,t1), (s2,t2) ... (sk,tk)
that needs to be routed in G.
X is well-linked in G.
Examples
s1
s2
t1
t2
s3
s4
t3
t4
s1
s2
t1
t2
s3
s4
t3
t4
Not a well-linked instance
A well-linked instance
Crossbars
H=(V,E) is a cross-bar with respect to an interface I µ V
if any matching on I can be routed using edge-disjoint
paths.
H
Ex: a complete graph is a cross-bar with I=V
Grids as Crossbars
First row is
s1
s2 s3
t1
t2 s4
t3 s5
t4 t 5
interface
Grids in Planar Graphs
Theorem [Robertson, Seymour, Thomas ’94]: If G is a
planar graph with a well-linked set of size k, then G has
a grid minor H of size W(k) as a subgraph.
v
Gv
Grid minor is a crossbar with congestion 2
[Kleinberg ’96]: uses it for half disjoint paths.
Gv
Routing pairs in X using H
H
X
Routing pairs in X using H
Sink
H
A Single-Source
Single-Sink Flow
Computation
X
Source
Routing pairs in X using H
Sink
H
Route X to I
X
Source
Routing pairs in X using H
H
Route X to I
X
Routing pairs in X using H
H
Route X to I
and use H for
pairing up
X
Several Technical Issues

H is smaller than X, so can pairs reach H?

What if X cannot reach H?

Can X reach interface of H without using edges of H?

Can H be found in polynomial time?
Routing Pairs to I
Claim: If some subset A of terminals can reach I,
then any subset A’ of terminals with |A’| · |A|/2
can reach I.
Use the fact that the terminals are well-linked.
Routing Pairs to I
V-S
S
I
A’
p edges
Routing Pairs to I
S
A’
A
A
V-S
I
p edges
If |S Å A| ¸ |A|/2, then p ¸ |A|/2 ¸ |A’| since A can reach B.
Routing Pairs to I
V-S
S
A’
A
A
I
p edges
If |(V-S) Å A| ¸ |A|/2, then p ¸ |A|/2 since terminals are
well-linked.
Routing Pairs to I
V-S
S
I
A’
p ¸ |A’| edges
Thus A’ can be routed to I.
We can choose any |A’|/2 pairs to be routed to I.
Summarizing ...
1. Solve the LP relaxation.
2. Use LP solution to decompose input instance into a
collection of well-linked instances.
3. Well-linked instances on planar graphs have a grid
crossbar. Use it to route many pairs.
Can Route the Entire Matching
 For EDP, suffices to route simply a constant
fraction of pairs in the EDP instance.
 Actually, we can route the entire matching with
O(1) congestion.
Decomposition into Well-linked Instances
G
G1
Xi is well-linked in Gi
G2
Gr
i |Xi| ¸ OPT/b
Example
s1
s2
t1
t2
s3
s4
t3
t4
s1
t1
s2
t2
s3
t3
s4
t4
Decomposition


b = O(log2 n) in general graphs.
b = O(log n) for planar graphs.
Decomposition based on LP solution.
EDP with Congestion in General Graphs
What is the integrality gap of multicommodity flow relaxation
when 2 paths can share an edge?
Is it more than a constant?
How Good is this LP?
tk
tk-1
ti
t3
t2
t1
sk sk-1
si
s3 s2 s1
Gap holds for planar graphs
EDP with Congestion in General Graphs
What is the integrality gap of multicommodity flow relaxation
when 2 paths can share an edge?
Is it more than a constant?
[Chuzhoy, K ’05]
A simple family of instances with gap roughly log1/c n for any
congestion c.
Shows that there is a superconstant gap between fractional flow
and 1/c-integral flow for any integer c.
Similar gap results hold even for All-or-Nothing Flow.
EDP with Congestion in General Graphs
EDP in general undirected graphs is logW(1/c) n-hard to
approximate when congestion of c is allowed.
[Andrews, Zhang’05], [Chuzhoy, K ’05],
[Guruswami, Talwar’05].
EDP with Congestion in General Graphs
EDP in general undirected graphs is logW(1/c) n-hard to
approximate when congestion of c is allowed.
[Andrews, Zhang’05], [Chuzhoy, K ’05],
[Guruswami, Talwar’05].
On the positive side, when congestion of c is allowed
O(n1/c)-approximation is known [Srinivasan’ 97],
[Baveja-Srinivasan’ 00], [Azar-Regev’ 01].
Summary

New algorithmic framework for routing problems.

Recent developments for undirected graphs



O(log2 n)-approximation for even degree planar graphs with no
congestion [Kleinberg ’05]
O(1)-approximation for planar graphs with O(1)-congestion
[Chekuri, K, Shepherd ’05]
Tight bound of Q(n1/2) for LP integrality gap in general graphs,
improves upon the previous upper bound of O(n^{2/3}).
[Chekuri, K, Shepherd
’05]
 Many open problems …
Thank You!
Routing Using the Grid Crossbar
Let k = |X| be the # of terminals.



Suppose we have a k/10 by k/10 grid crossbar H.
Route terminals in X to interface vertices in H:
a
single source-single sink max flow computation.
If max flow ¸ k/100, we can route in an edge-disjoint
manner at least k/100 terminals to the interface and pair
them using the grid.
EDP on Line Networks
si
ti
Independent set in interval graphs.
Advantage of Well-linkedness
LP value does not depend on input pairing M.
s1
s2
t1
t2
s3
s4
t3
t4
Claim: If X is well-linked, then for any pairing on X, LP
value is W(|X|/log |X|).
We have symmetry w.r.t. to every pairing.