Approximation Algorithms for
Single and Multi-Commodity
Connected Facility Location
Fabrizio Grandoni & Thomas Rothvoß
Department of Mathematics, M.I.T.
IPCO 2011
Outline
1.
Better approximation algorithm
for Connected Facility Location
Outline
1.
2.
Better approximation algorithm
for Connected Facility Location
First O(1)-apx for Multi-Commodity
Connected Facility Location
Outline
1.
2.
3.
Better approximation algorithm
for Connected Facility Location
First O(1)-apx for Multi-Commodity
Connected Facility Location
Improved hardness results for several problems
Part 1: Connected Facility Location
Given:
◮ graph G = (V, E), with
metric distances c : E → Q+
Part 1: Connected Facility Location
Given:
◮ graph G = (V, E), with
metric distances c : E → Q+
◮
a set of clients C ⊆ V
clients
Part 1: Connected Facility Location
Given:
◮ graph G = (V, E), with
metric distances c : E → Q+
◮
a set of clients C ⊆ V
◮
facilities F ⊆ V , with
opening cost o : F → Q+
facility
clients
Part 1: Connected Facility Location
Given:
◮ graph G = (V, E), with
metric distances c : E → Q+
◮
a set of clients C ⊆ V
◮
facilities F ⊆ V , with
opening cost o : F → Q+
◮
a parameter M ≥ 1
facility
clients
Part 1: Connected Facility Location
Given:
◮ graph G = (V, E), with
metric distances c : E → Q+
◮
a set of clients C ⊆ V
◮
facilities F ⊆ V , with
opening cost o : F → Q+
◮
a parameter M ≥ 1
facility
Goal:
◮ open facilities F ′ ⊆ F
minimizing
X
o(f ) +
{z
c(j, F ′ )
|
{z
j∈C
f ∈F ′
|
X
}
opening cost
}
connection cost
clients
Part 1: Connected Facility Location
Given:
◮ graph G = (V, E), with
metric distances c : E → Q+
◮
a set of clients C ⊆ V
◮
facilities F ⊆ V , with
opening cost o : F → Q+
◮
a parameter M ≥ 1
facility
clients
Goal:
◮ open facilities F ′ ⊆ F
◮ find Steiner tree T spanning opened facilities
minimizing
X
X
X
c(j, F ′ ) + M
c(e)
o(f ) +
j∈C
f ∈F ′
|
{z
}
opening cost
|
e∈T
{z
}
connection cost
|
{z
}
Steiner cost
Previous Results for Connected Facility
Location
◮
APX-hard (reduction from Steiner Tree)
◮
10.66-apx based on LP-rounding [Gupta et al ’01].
◮
8.55-apx primal-dual algorithm
[Swamy, Kumar ’02].
◮
3.92-apx based on random-sampling
[Eisenbrand, Grandoni, R., Schäfer ’08]
Previous Results for Connected Facility
Location
◮
APX-hard (reduction from Steiner Tree)
◮
10.66-apx based on LP-rounding [Gupta et al ’01].
◮
8.55-apx primal-dual algorithm
[Swamy, Kumar ’02].
◮
3.92-apx based on random-sampling
[Eisenbrand, Grandoni, R., Schäfer ’08]
Our result
Simple 3.19-approximation algorithm.
Previous Results for Connected Facility
Location
◮
APX-hard (reduction from Steiner Tree)
◮
10.66-apx based on LP-rounding [Gupta et al ’01].
◮
8.55-apx primal-dual algorithm
[Swamy, Kumar ’02].
◮
3.92-apx based on random-sampling
[Eisenbrand, Grandoni, R., Schäfer ’08]
Our result
Simple 3.19-approximation algorithm.
Use existing algorithms for subproblems:
◮
Facility Location: 1.5-apx [Byrka ’07]
◮
Steiner Tree: 1.39-apx [Byrka,Grandoni,R.,Sanità 10]
Where is the difficulty?
Algorithm:
opening cost 0
Where is the difficulty?
Algorithm:
(1) Just solve Facility Location
opening cost 0
Where is the difficulty?
Algorithm:
(1) Just solve Facility Location
(2) Run Steiner Tree algorithm
opening cost 0
pay tree M times!
Where is the difficulty?
Algorithm:
(1) Just solve Facility Location
!
k
r
o
w
’t
(2) Run Steiner Tree algorithm
opening cost 0
n
s
e
o
D
pay tree M times!
The CFL algorithm
The CFL algorithm
(1) Guess facility r from OP T .
r
The CFL algorithm
(1) Guess facility r from OP T .
(2) Sample each client with prob
r
1
M
→ C′
The CFL algorithm
(1) Guess facility r from OP T .
1
→ C′
(2) Sample each client with prob M
′
(3) Compute 1.5-apx F for Facility Location instance
o′ (f ) := o(f ) + M · c(f, nearest node in C ′ ∪ {r})
r
o(f )+ penalty
The CFL algorithm
(1) Guess facility r from OP T .
1
→ C′
(2) Sample each client with prob M
′
(3) Compute 1.5-apx F for Facility Location instance
o′ (f ) := o(f ) + M · c(f, nearest node in C ′ ∪ {r})
r
The CFL algorithm
(1) Guess facility r from OP T .
1
→ C′
(2) Sample each client with prob M
′
(3) Compute 1.5-apx F for Facility Location instance
o′ (f ) := o(f ) + M · c(f, nearest node in C ′ ∪ {r})
(4) Compute 1.39-apx Steiner Tree T ′ on F ′ ∪ {r}
r
The CFL algorithm
(1) Guess facility r from OP T .
1
→ C′
(2) Sample each client with prob M
′
(3) Compute 1.5-apx F for Facility Location instance
o′ (f ) := o(f ) + M · c(f, nearest node in C ′ ∪ {r})
(4) Compute 1.39-apx Steiner Tree T ′ on F ′ ∪ {r}
r
◮
Observe:




cost of apx
cost of apx
E[AP X] ≤ E  Facility Location sol. +E  Steiner Tree 
+penalties
The CFL algorithm
(1) Guess facility r from OP T .
1
→ C′
(2) Sample each client with prob M
′
(3) Compute 1.5-apx F for Facility Location instance
o′ (f ) := o(f ) + M · c(f, nearest node in C ′ ∪ {r})
(4) Compute 1.39-apx Steiner Tree T ′ on F ′ ∪ {r}
r
◮
Observe:




cost of apx
cost of apx
E[AP X] ≤ E  Facility Location sol. +E  Steiner Tree 
+penalties
on C ′ ∪ {r}
Analysis: Steiner Cost
Notation:
OP T =
O∗
|{z}
opening cost
+
S∗
|{z}
Steiner tree cost
+
C∗
|{z}
connection cost
Analysis: Steiner Cost
Notation:
OP T =
O∗
|{z}
opening cost
+
S∗
|{z}
Steiner tree cost
+
C∗
|{z}
connection cost
Lemma
E[M · apx Steiner Tree on C ′ ∪ {r}] ≤ 1.39 · (S ∗ + C ∗ )
r
Analysis: Steiner Cost
Notation:
OP T =
O∗
|{z}
+
opening cost
S∗
|{z}
Steiner tree cost
+
C∗
|{z}
connection cost
Lemma
E[M · apx Steiner Tree on C ′ ∪ {r}] ≤ 1.39 · (S ∗ + C ∗ )
r
S∗
|{z}
Steiner tree
in OP T
Analysis: Steiner Cost
Notation:
OP T =
O∗
|{z}
+
opening cost
S∗
|{z}
Steiner tree cost
+
C∗
|{z}
connection cost
Lemma
E[M · apx Steiner Tree on C ′ ∪ {r}] ≤ 1.39 · (S ∗ + C ∗ )
r
S∗
|{z}
Steiner tree
in OP T
+
1
· M · C∗
M
{z
}
|
need path in C ∗
with prob. 1/M
Analysis: Steiner Cost
Notation:
OP T =
O∗
|{z}
+
opening cost
S∗
|{z}
Steiner tree cost
+
C∗
|{z}
connection cost
Lemma
E[M · apx Steiner Tree on C ′ ∪ {r}] ≤ 1.39 · (S ∗ + C ∗ )
r
1.39 ·
S∗
|{z}
Steiner tree
in OP T
+
1
· M · C∗
}
|M {z
need path in C ∗
with prob. 1/M
Core detouring theorem
Core Detouring Theorem [EGSR ’08]
Given a spanning tree T with root r ∈ T and distinguished
terminals D ⊆ V . Sample any terminal in D with prob. p ∈]0, 1]
E
hX
v∈D
i 0.81
· c(T )
dist(v, sampled node ∪ {r}) ≤
p
r
T
Core detouring theorem
Core Detouring Theorem [EGSR ’08]
Given a spanning tree T with root r ∈ T and distinguished
terminals D ⊆ V . Sample any terminal in D with prob. p ∈]0, 1]
E
hX
v∈D
i 0.81
· c(T )
dist(v, sampled node ∪ {r}) ≤
p
r
r
r
T
r
Core detouring theorem
Core Detouring Theorem [EGSR ’08]
Given a spanning tree T with root r ∈ T and distinguished
terminals D ⊆ V . Sample any terminal in D with prob. p ∈]0, 1]
E
hX
v∈D
i 0.81
· c(T )
dist(v, sampled node ∪ {r}) ≤
p
r
r
r
T
r
Analysis: Facility Location cost
Theorem
E[apx Facility Location cost] ≤ 1.5 · (O ∗ + 2C ∗ + 0.81S ∗ )
r
Analysis: Facility Location cost
Theorem
E[apx Facility Location cost] ≤ 1.5 · (O ∗ + 2C ∗ + 0.81S ∗ )
◮
There is a good sol: F := facilities in OP T serving C ′ ∪ {r}
r
Analysis: Facility Location cost
Theorem
E[apx Facility Location cost] ≤ 1.5 · (O ∗ + 2C ∗ + 0.81S ∗ )
◮
◮
′
There
is a good sol:
F := facilities in OP T serving C ∪ {r}
1
opening cost
E
≤ |{z}
O∗ +
· M · C∗
+ penalties
M
{z
}
|
orig. cost
r
expected penalty
Analysis: Facility Location cost
Theorem
E[apx Facility Location cost] ≤ 1.5 · (O ∗ + 2C ∗ + 0.81S ∗ )
◮
◮
′
There
is a good sol:
F := facilities in OP T serving C ∪ {r}
1
opening cost
E
≤ |{z}
O∗ +
· M · C∗
+ penalties
M
{z
}
|
orig. cost
◮
E[connection cost] ≤ C ∗ +
r
expected penalty
Analysis: Facility Location cost
Theorem
E[apx Facility Location cost] ≤ 1.5 · (O ∗ + 2C ∗ + 0.81S ∗ )
◮
◮
′
There
is a good sol:
F := facilities in OP T serving C ∪ {r}
1
opening cost
E
≤ |{z}
O∗ +
· M · C∗
+ penalties
M
{z
}
|
orig. cost
◮
expected penalty
0.81 S ∗
E[connection cost] ≤ C ∗ +
·
1/M |{z}
M
tree
Use Core Detouring Theorem with T = S ∗ and p :=
r
1
M.
Finishing the analysis for CFL
Conclusion:
E[AP X]
≤
1.39 · (S ∗ + C ∗ ) + 1.5 · (O ∗ + 2C ∗ + 0.81S ∗ )
Finishing the analysis for CFL
Conclusion:
E[AP X]
≤
1.39 · (S ∗ + C ∗ ) + 1.5 · (O ∗ + 2C ∗ + 0.81S ∗ )
≤
1.5 · O ∗ + 2.7 · S ∗ + 4.39 · C ∗
Finishing the analysis for CFL
Conclusion:
E[AP X]
≤
1.39 · (S ∗ + C ∗ ) + 1.5 · (O ∗ + 2C ∗ + 0.81S ∗ )
≤
1.5 · O ∗ + 2.7 · S ∗ + 4.39 · C ∗
Improvement to 3.19:
◮
Adapting the sampling probability
◮
Using a bi-factor Facility Location algorithm
Part 2: Multi-Commodity Connected Facility
Location
s3
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
◮
parameter M ≥ 1
s6
s5
r1
s1
r2
r6
s2
s4
r4
r5
r3
Part 2: Multi-Commodity Connected Facility
Location
s3
s6
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
parameter M ≥ 1
Goal: Find
◮ Facilities F ′ ⊆ F
o(f )
{z
s1
r2
s2
s4
r4
r5
r3
f ∈F ′
|
r1
r6
◮
X
s5
}
opening cost
Part 2: Multi-Commodity Connected Facility
Location
s3
s6
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
parameter M ≥ 1
Goal: Find
◮ Facilities F ′ ⊆ F
◮ Forest T
X
o(f ) + M · c(T )
| {z }
f ∈F ′
forest cost
| {z }
◮
opening cost
s5
r1
s1
r2
r6
s2
s4
r4
r5
r3
Part 2: Multi-Commodity Connected Facility
Location
s3
s6
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
s5
r1
s1
r2
r6
s2
s4
r4
r5
parameter M ≥ 1
Goal: Find
◮ Facilities F ′ ⊆ F
◮ Forest T
k
X
X
distF ′ ,T (si , ri )
o(f ) + M · c(T ) +
| {z }
|
{z
}
i=1
f ∈F ′
forest cost
0 distance between
| {z }
opened & connected
◮
opening cost
facilities
r3
Part 2: Multi-Commodity Connected Facility
Location
s3
s6
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
s5
r1
s1
r2
r6
s2
s4
r4
r5
parameter M ≥ 1
Goal: Find
◮ Facilities F ′ ⊆ F
◮ Forest T
k
X
X
distF ′ ,T (si , ri )
o(f ) + M · c(T ) +
| {z }
|
{z
}
i=1
f ∈F ′
forest cost
0 distance between
| {z }
opened & connected
◮
opening cost
facilities
r3
Part 2: Multi-Commodity Connected Facility
Location
s3
s6
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
s5
r1
s1
r2
r6
s2
s4
r4
r5
parameter M ≥ 1
Goal: Find
◮ Facilities F ′ ⊆ F
◮ Forest T
k
X
X
distF ′ ,T (si , ri )
o(f ) + M · c(T ) +
| {z }
|
{z
}
i=1
f ∈F ′
forest cost
0 distance between
| {z }
opened & connected
◮
opening cost
facilities
r3
Part 2: Multi-Commodity Connected Facility
Location
s3
s6
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
s5
r1
s1
r2
r6
s2
s4
r4
r5
parameter M ≥ 1
Goal: Find
◮ Facilities F ′ ⊆ F
◮ Forest T
k
X
X
distF ′ ,T (si , ri )
o(f ) + M · c(T ) +
| {z }
|
{z
}
i=1
f ∈F ′
forest cost
0 distance between
| {z }
opened & connected
◮
opening cost
facilities
r3
Part 2: Multi-Commodity Connected Facility
Location
s3
s6
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
s5
r1
s1
r2
r6
s2
s4
r4
r5
parameter M ≥ 1
Goal: Find
◮ Facilities F ′ ⊆ F
◮ Forest T
k
X
X
distF ′ ,T (si , ri )
o(f ) + M · c(T ) +
| {z }
|
{z
}
i=1
f ∈F ′
forest cost
0 distance between
| {z }
opened & connected
◮
opening cost
facilities
r3
Part 2: Multi-Commodity Connected Facility
Location
s3
s6
Input:
◮
Undirected graph G = (V, E),
metric distances c : E → Q+
◮
source-sink pairs
(s1 , r1 ), . . . , (sk , rk )
◮
a set of facilities F ⊆ V with
opening costs o : F → Q+
s5
r1
s1
r2
r6
s2
s4
r4
r5
parameter M ≥ 1
Goal: Find
◮ Facilities F ′ ⊆ F
◮ Forest T
k
X
X
distF ′ ,T (si , ri )
o(f ) + M · c(T ) +
| {z }
|
{z
}
i=1
f ∈F ′
forest cost
0 distance between
| {z }
opened & connected
◮
opening cost
facilities
r3
Our result
Our result
Simple 16.2-approximation algorithm for Multi-Commodity
Connected Facility Location.
Ingredients:
◮
◮
Random-sampling
Use algorithms for
◮
◮
Price-Collecting Facility Location
Steiner Forest
Part 3: Improved hardness results
Steiner Tree
Facility Location
generalizes
Single-Sink
Rent-or-Buy
Connected
Facility Location
Single-Sink
Buy-at-Bulk
Virtual Private
Network
Part 3: Improved hardness results
Steiner Tree
1.01
Facility Location
1.46
generalizes
Single-Sink
Rent-or-Buy
1.01
Single-Sink
Buy-at-Bulk
1.01
Connected
Facility Location
1.46
Virtual Private
Network
1.01
Part 3: Improved hardness results
Steiner Tree
1.01
Facility Location
1.46
generalizes
Single-Sink
Rent-or-Buy
8
1.01
1.2
Single-Sink
Buy-at-Bulk
8
1.01
1.2
Connected
Facility Location
1.46
Virtual Private
Network
8
1.01
1.2
Part 3: Improved hardness results
Steiner Tree
1.01
Facility Location
1.46
generalizes
Single-Sink
o(f ) = 0
Rent-or-Buy
∀f ∈ V
8
1.01
1.2
Single-Sink
Buy-at-Bulk
8
1.01
1.2
Connected
Facility Location
1.46
Virtual Private
Network
8
1.01
1.2
The reduction (1)
◮
Reduce Set Cover to SROB
elements
sets
b
The reduction (1)
◮
Reduce Set Cover to SROB
clients/elements
c(e) ∈ {1, 2}
M=
facilities/sets
r
0.27n
OP TSC
The reduction (1)
◮
Reduce Set Cover to SROB
clients/elements
c(e) ∈ {1, 2}
2
1
M=
facilities/sets
r
0.27n
OP TSC
The reduction (1)
◮
Reduce Set Cover to SROB
remove!
clients/elements
c(e) ∈ {1, 2}
2
1
facilities/sets
M=
buy!
0.27n
OP TSC
r
(1) WHILE not all elements covered DO
(2) Compute 1.27-apx SROB sol
(3) Buy facilities/sets in sol. & remove covered elements
The reduction (2)
◮
Use idea from [Guha & Khuller ’99]
# clients at dist 1
n
# fac
0.27 OP
TSC n + 0.3667n
# facilities in apx. sol
1 · OP TSC 2 · OP TSC
The reduction (2)
◮
Use idea from [Guha & Khuller ’99]
# clients at dist 1
n
apx SROB sol.
# fac
0.27 OP
TSC n + 0.3667n
b
# facilities in apx. sol
1 · OP TSC 2 · OP TSC
The reduction (2)
◮
Use idea from [Guha & Khuller ’99]
# clients at dist 1
n
apx SROB sol.
# fac
0.27 OP
TSC n + 0.3667n
b
# facilities in apx. sol
1 · OP TSC 2 · OP TSC
# needed sets
≤ [. . . some calc . . .] ≤ 0.999 ln(n) · OP TSC
The reduction (2)
◮
Use idea from [Guha & Khuller ’99]
# clients at dist 1
n
apx SROB sol.
# fac
0.27 OP
TSC n + 0.3667n
b
# facilities in apx. sol
1 · OP TSC 2 · OP TSC
# needed sets
◮
≤ [. . . some calc . . .] ≤ 0.999 ln(n) · OP TSC
Contradiction!
Theorem (Feige ’98)
Unless NP ⊆ DTIME(nO(log log n) ), there is no
(1 − ε) · ln(n)-apx for Set Cover.
The end
Thanks for your attention