Approximation Algorithms for
Graph Routing Problems
Julia Chuzhoy
Toyota Technological Institute at
Chicago
Routing Problems
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible; minimize
edge congestion.
Routing Problems
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible; minimize
edge congestion.
Routing Problems
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible; minimize
edge congestion.
Edge congestion: 2
Routing Problems
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible; minimize
edge congestion.
n – number of graph vertices
m – number of edges
k – number of demand pairs
terminals – vertices participating
in the demand pairs
Routing Problems
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible; minimize
edge congestion.
• 3 pairs with congestion 2
Routing Problems
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible; minimize
edge congestion.
• 3 pairs with congestion 2
• 2 pairs with congestion 1
Congestion Minimization
Congestion Minimization
• Route all demand pairs
• Minimize maximum edge congestion
solution value: 2
LP-relaxation
Rounding Algorithm [Raghavan, Thompson ‘87]
• Every pair (si,ti) chooses a path P w. probability f(P)
• Max edge congestion O(clog n/log log n) w.h.p.
Congestion Minimization
•
-approximation [Raghavan,
Thompson ‘87]
• Directed graphs:
-hard to
approximate [Andrews, Zhang ‘06], [C, Guruswami,
Khanna, Talwar ‘07]
• Undirected graphs:
-hard to
approximate [Andrews, Zhang ‘07]
Open Problem 1
Can we close the gap for undirected graphs?
• What is the integrality gap of the LP?
• What if we only need to connect a constant
fraction of the demand pairs?
Edge-Disjoint Paths Problem
• No congestion allowed
• Route maximum number of the demand pairs
Solution value: 2
Edge Disjoint Paths (EDP)
• For directed graphs NP-hard even with only two
demand pairs [Fortune, Hopcroft, Wyllie '80]
• When k is constant, can be solved efficiently in
undirected graphs [Robertson, Seymour ‘90]
• NP-hard when k is part of input [Karp ’72]
LP-relaxation
Rounding Algorithm [Kolliopoulos, Stein ‘98]
• Find shortest path P with non-zero flow,
connecting any demand pair.
• Add P to the solution and delete all flow that
uses edges of P.
-approximation
Analysis
• If the length of P is less than
flow is deleted.
• If the length of P is more than
flow remains.
– at most
– at most
Can We Do Better?
Directed graphs: EDP is
-hard to
approximate for any [Guruswami, Khanna,
Rajaraman, Shepherd, Yannakakis ‘99]
Undirected graphs:
•
-approximation algorithm [Chekuri, Khanna,
Shepherd ’06].
•
-hardness of approximation for
any[Andrews, Zhang ‘05], [Andrews, C, Guruswami, Khanna,
Talwar, Zhang ’10]
• LP integrality gap:
‘93]
[Garg, Vazirani, Yannakakis
Integrality Gap Example
[Garg, Vazirani, Yannakakis ‘93]
t1
t2
…
tk
s1 s2 …
sk
• fractional: k/2
• integral: 1
• k=
A Brick-Wall Graph
s1
tk
…
s2
…
t2
sk
t1
Open Problem 2
Close the gap for undirected EDP
• planar graphs?
– constant-factor approximation with congestion 2
[Chekuri, Khanna, Shepherd ‘04], [Chekuri, Khanna,
Shepherd ‘06], [Seguin-Charbonneau, Shepherd ‘11]
– O(log n)-approximation for Eulerian or 4-edge
connected planar graphs [Kawarabayashi, Kobayashi
’10]
– constant approximation for grid-like graphs
[Kleinberg, Tardos ’95]
Open Problem 2
Close the gap for undirected EDP
• planar graphs?
• better algorithms for brick-wall graphs?
Positive Results
Moderately connected graphs: If global min-cut
is Ω(log5n), there is a polylog-approximation [Rao,
Zhou ‘10]
Expander graphs
In a strong enough constant-degree expander, any
demand set on
vertices can be routed
on edge-disjoint paths [Frieze ‘00]
…
Edge Disjoint Paths (EDP)
Congestion Minimization
Route maximum number of
pairs on edge-disjoint paths
Route all pairs; minimize
congestion
•
-approximation
• matching integrality gap
•
•
•
-hardness
-approximation
-hardness
EDP with Congestion (EDPwC)
A factor- approximation algorithm with congestion c
routes
. demand pairs with congestion at most c.
optimum number of pairs
with no congestion allowed
EDPwC
• Congestion O(log n/log log n): constant
approximation [Raghavan, Thompson ’87]
•
-approximation with congestion c [Azar, Regev
’01], [Baveja, Srinivasan ’00], [Kolliopoulos, Stein ‘04]
Directed graphs:
-hardness for any congestion
c [C, Guruswami, Khanna, Talwar ’06]
EDPwC
• Congestion O(log n/log log n): constant
approximation [Raghavan, Thompson ’87]
•
-approximation with congestion c [Azar, Regev
’01], [Baveja, Srinivasan ’00], [Kolliopoulos, Stein ‘04]
• polylog(n)-approximation with congestion
poly(log log n) [Andrews ‘10]
• Congestion 2:
-approximation [Kawarabayashi,
Kobayashi ’11]
• polylog(k)-approximation with congestion 14 [C, ‘11]
• polylog(k)-approximation with congestion 2 [C, Li, ‘12]
Integrality Gaps for EDPwC
• Congestion 1: integrality gap
[Garg, Vazirani,
Yannakakis ‘93]
• Congestion c: integrality gap
[Andrews, C, Guruswami, Khanna, Talwar, Zhang ‘10]
•
-hard to approximate
with
t1
congestion c for any
t2
…
tk
s 1 s2 … s k
Edge Disjoint Paths (EDP)
Congestion Minimization
Route maximum number of
pairs on edge-disjoint paths
Route all pairs; minimize
congestion
•
-approximation
• matching integrality gap
•
•
•
-hardness
-approximation
-hardness
EDP with Congestion (EDPwC)
• polylog(k)-approximation with congestion 2
•
-hardness with congestion c
Another View: Reducing Congestion
• Suppose we have a “bad” solution, where X
pairs are routed with congestion C.
• Then we can route X/(C polylog k) pairs with
congestion 2!
• But if we want congestion 1, may only be able
to route
pairs, even if C=2.
A Polylogarithmic Approximation with
Constant Congestion
Well-Linkedness
[Robertson,Seymour], [Chekuri, Khanna, Shepherd], [Raecke]
Graph G is well-linked for the set T of
terminals, iff for any partition (A,B) of V(G),
Well-Linkedness
[Robertson,Seymour], [Chekuri, Khanna, Shepherd], [Raecke]
Graph G is well-linked for the set T of
terminals, iff for any partition (A,B) of V(G),
Pre-Processing
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Theorem [Chekuri, Khanna Shepherd ‘04]
Can efficiently partition G into disjoint
subgraphs
G1,…,toGsolve
that:
“Enough”
the
problem on wellr, such
linked
instances. Gi the
• For each induced
sub-problem
terminals are well-linked
• Total fractional solution value for all induced
sub-problems is
High-Level Plan [CKS ‘04]
1. Embed an expander on a subset of terminals
into G.
Crossbar
– expander vertices
– expander edges
terminals
paths in G
2. Route a subset of the
demand
pairs in the
Embedding
congestion:
max load on any edge of G
expander
Goal
Embed an expander over a subset of terminals
into G.
• Include polylog(k)-fraction of the terminals
• congestion 2
Cut-Matching Game [Khandekar, Rao,
Vazirani ’06]
Cut Player: wants to build an expander
Matching Player: wants to delay its construction
There is a strategy for cut player, s.t.
after O(log2n) iterations, we get an
expander!
Embedding Expander into Graph
Embedding Expander into Graph
After O(log2k) iterations, we get an expander embedded
into G.
Problem: congestion Ω(log2k)
Solution?
Idea [Rao Zhou ‘06]:
• Split G into graphs G1,…,Gh using algorithm of [Karger
’93]
–
–
–
–
V(Gi)=V(G) for all i
Every edge of G belongs to at most one Gi
Each Gi well-linked for the terminals
h=O(log2k)
• Run the cut-matching game. Use Gi to route flow in
iteration i.
Problem: Can only do it if min-cut in G is Ω(log5n)
• [Andrews ‘10] adapted this to general graphs, with
congestion poly(log log n)
Getting a Constant Congestion
Embedding an Expander into G
Expander vertex
connected component in G
containing the terminal
Expander edge
path connecting some pair of
vertices in the two components
An edge of G belongs to at most 2 of the
components/paths.
Embedding an Expander into G
Routing on vertex-disjoint paths in X gives a
congestion-2 routing in G!
Embedding an Expander into G
Expander vertex
connected component in G
containing the terminal
Expander edge
path connecting some pair of
vertices in the two components
An edge of G belongs to at most 2 of the
components/paths.
Embedding an Expander into G
Families of Good Vertex Sets
Good Vertex Subset
S is a good vertex subset iff:
• S contains no terminals
• k/polylog k red edges
• S is well-linked for the
red edges
A Good Family of Vertex Subsets
• O(log2k) disjoint good vertex subsets
• k/polylog k trees
− each edge of G participates in at most 2 trees
− Each tree Ti spans a distinct terminal ti and a
distinct red edge adjacent to Sj for each j.
Embedding an Expander
Embedding an Expander
Expander vertex
the tree spanning the terminal
Expander edges: via the cut-matching game of [KRV]
Embedding an Expander
Embedding an Expander
After O(log2k) iterations, we obtain
an expander embedded into G
with congestion 2.…
Algorithm for EDPwC
Find a good family of vertex subsets
Embed an expander into G
Find vertex-disjoint routing in the expander
Transform into routing in G
Open Problem 3
• A cleaner algorithm for polylog(k)
approximation with congestion 2? Better power
of polylog?
• What if we need to route almost all demand
pairs (a constant fraction)?
Routing on Vertex-Disjoint Paths
• Generalizes EDP
• Connections to Graph Minor theory
No congestion:
•
-approximation [Kolliopoulos, Stein ‘98]
•
-lower bound on LP integrality gap
[Garg, Vazirani, Yannakakis ‘97]
Routing with congestion:
• O(poly log k)-approximation with constant
congestion [Chekuri, Ene ‘12]
Open Problem 4
• Better algorithms/lower bounds for VertexDisjoint Paths?
Open Problem 4
• Better algorithms/lower bounds for VertexDisjoint Paths?
• Vertex-disjoint paths in grid graphs
Open Problem 4
• Better algorithms/lower bounds for VertexDisjoint Paths?
• Vertex-disjoint paths in grid graphs
s1
s3
s2
t1
t2
t3
Thank you!