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FTFP Results Related Work Techniques Algorithms End
Approximation Algorithms for the Fault-TolerantFacility Placement Problem
Li Yan
Computer ScienceUniversity of California Riverside
06/10/2013
1 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Outline
1 The FTFP Problem
2 Results in Dissertation
3 Related Work
4 Techniques
5 Approximation Algorithms
6 Summary
2 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Outline
1 The FTFP Problem
2 Results in Dissertation
3 Related Work
4 Techniques
5 Approximation Algorithms
6 Summary
3 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Fault-Tolerant Facility Placement Problem (FTFP)
3
2
1
f3 = 4
f2 = 6
f1 = 3
4
3
2
1
r4 = 2
r3 = 1
r2 = 1
r1 = 41
2
1
3
2
4
site client
demandcost toopen facility
connectioncost
4 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Fault-Tolerant Facility Placement Problem (FTFP)
3
2
1
f3 = 4
f2 = 6
f1 = 3
4
3
2
1
r4 = 2
r3 = 1
r2 = 1
r1 = 41
2
1
3
2
4
site client
demandcost toopen facility
connectioncost
4 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Feasible Integral Solution
3
2
1
f3 = 4
f2 = 6
f1 = 3
4
3
2
1
r4 = 2
r3 = 1
r2 = 1
r1 = 41
2
13
24
Instance
3
2
1
4
3
2
1
Solution
Cost
2f1 + f2 + 3f3 + d11 + d12 + 2d14 + d23 + 3d31 = 38
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FTFP Results Related Work Techniques Algorithms End
Feasible Integral Solution
3
2
1
f3 = 4
f2 = 6
f1 = 3
4
3
2
1
r4 = 2
r3 = 1
r2 = 1
r1 = 41
2
13
24
Instance
3
2
1
4
3
2
1
Solution
Cost
2f1 + f2 + 3f3 + d11 + d12 + 2d14 + d23 + 3d31 = 38
5 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Feasible Integral Solution
3
2
1
f3 = 4
f2 = 6
f1 = 3
4
3
2
1
r4 = 2
r3 = 1
r2 = 1
r1 = 41
2
13
24
Instance
3
2
1
4
3
2
1
Solution
Cost
2f1 + f2 + 3f3 + d11 + d12 + 2d14 + d23 + 3d31 = 38
5 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Metric Distances: Triangle Inequality
i1
i2
j1
j2
Triangle Inequality
d(i1, j2) ≤d(i1, j1) + d(i2, j1) + d(i2, j2)
Needed when estimatingdistances...
6 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Outline
1 The FTFP Problem
2 Results in Dissertation
3 Related Work
4 Techniques
5 Approximation Algorithms
6 Summary
7 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Hardness
How hard is FTFP?
FTFP is NP-hard
FTFP is MaxSNP-hard
Best ratio ≥ 1.463 unless P = NP
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FTFP Results Related Work Techniques Algorithms End
Hardness
How hard is FTFP?
FTFP is NP-hard
FTFP is MaxSNP-hard
Best ratio ≥ 1.463 unless P = NP
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FTFP Results Related Work Techniques Algorithms End
Hardness
How hard is FTFP?
FTFP is NP-hard
FTFP is MaxSNP-hard
Best ratio ≥ 1.463 unless P = NP
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FTFP Results Related Work Techniques Algorithms End
NP-hard Optimization Problem
Integer Program
Linear Program (LP) with Fractional Solution
LP-rounding Primal-dual
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FTFP Results Related Work Techniques Algorithms End
Results Highlight
LP-rounding: 1.575-approximation
LP-rounding: asymptotic ratio of 1 when all demandslarge
Primal-dual: Hn-approximation
Primal-dual: Example of Ω(log n/ log log n) for dual-fitting
10 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Relation between Problems
FTFP rj ≥ 1 <∞ facility per siteUFL rj = 1 ≤ 1 facility per siteFTFL rj ≥ 1 ≤ 1 facility per site
UFL FTFP FTFL
LP-roundingUFLFTFP
1.575
FTFL 1.7245
Primal-dualUFL 1.52FTFPFTFL
O(log n)
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FTFP Results Related Work Techniques Algorithms End
Relation between Problems
FTFP rj ≥ 1 <∞ facility per siteUFL rj = 1 ≤ 1 facility per siteFTFL rj ≥ 1 ≤ 1 facility per site
UFL FTFP FTFL
LP-roundingUFLFTFP
1.575
FTFL 1.7245
Primal-dualUFL 1.52FTFPFTFL
O(log n)
11 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Relation between Problems
FTFP rj ≥ 1 <∞ facility per siteUFL rj = 1 ≤ 1 facility per siteFTFL rj ≥ 1 ≤ 1 facility per site
UFL FTFP FTFL
LP-roundingUFLFTFP
1.575
FTFL 1.7245
Primal-dualUFL 1.52FTFPFTFL
O(log n)
11 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End
Relation between Problems
FTFP rj ≥ 1 <∞ facility per siteUFL rj = 1 ≤ 1 facility per siteFTFL rj ≥ 1 ≤ 1 facility per site
UFL FTFP FTFL
LP-roundingUFLFTFP
1.575
FTFL 1.7245
Primal-dualUFL 1.52FTFPFTFL
O(log n)
11 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End UFL FTFL
Outline
1 The FTFP Problem
2 Results in Dissertation
3 Related Work
4 Techniques
5 Approximation Algorithms
6 Summary
12 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End UFL FTFL
Related Work for UFL
Approximation Results for UFL
Shmoys, Tardos and Aardal 1997 3.16 LP-roundingChudak 1998 1.736 LP-roundingSviridenko 2002 1.58 LP-roundingJain and Vazirani 2001 3 primal-dualJain et al. 2002 1.61 greedyMahdian et al. 2002 1.52 greedyArya et al. 2004 3 local searchByrka 2007 1.5 hybridLi 2011 1.488 hybrid
Lower Bound
Guha and Khuller 1998 1.463
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FTFP Results Related Work Techniques Algorithms End UFL FTFL
Related Work for FTFL
Approximation Algorithms for FTFL
Jain and Vazirani 2000 3 ln maxj rj primal-dualGuha et al. 2001 4 LP-roundingSwamy, Shmoys 2008 2.076 LP-roundingByrka et al. 2010 1.7245 LP-rounding
No primal-dual algorithms for FTFL with constant ratio.
14 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End UFL FTFL
Work on FTFP (Dissertation Topic)
Approximation Algorithms for FTFP
Xu and Shen 2009 Introduced FTFPLiao and Shen 2011 1.861 Dual-fitting (for special case)Yan and Chrobak 2011 3.16 LP-roundingYan and Chrobak 2012 1.575 LP-roundingYan and Chrobak preliminary results Dual-fitting (for general case)
15 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Outline
1 The FTFP Problem
2 Results in Dissertation
3 Related Work
4 Techniques
5 Approximation Algorithms
6 Summary
16 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Algorithm for FTFP — LP
yi = number of facilities open at site i ∈ F
xij = number of connections from client j ∈ C to site i ∈ F
(Primal) minimize∑
fiyi +∑
dijxij
subject to yi − xij ≥ 0 ∀i , j∑xij ≥ rj ∀j
xij ≥ 0, yi ≥ 0 ∀i , j
(Dual) maximize∑
rj αj
subject to∑
βij ≤ fi ∀i
αj − βij ≤ dij ∀i , jαj ≥ 0, βij ≥ 0 ∀i , j
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Algorithm for FTFP — Demand Reduction
LP x∗, y∗
x , yrj =
∑i xij
x , yrj =
∑i xij
ρ-approximationalgorithm
x , yx + xy + y
ρ-approximatesolution
LP-solver
round downto integer
fractional part
demand reduction
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Algorithm for FTFP — Adaptive Partitioning
LP x∗, y∗
x , yrj =
∑i xij
x , yrj =
∑i xij
x , yx , yx + xy + y
ρ-approximatesolution
LP-solver
round downto integer
fractional part
adaptivepartitioning
LP-rounding
demand reduction
19 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Techniques
Demand Reduction
Reduce all rj to polynomial values (to ensurepolynomial time of rounding)
ρ-approx for reduced instance ⇒ ρ-approx for originalinstance
Adaptive Partitioning
Split sites into facilities and clients into unit demands
Split associated fractional values
Properties ensure rounding similar to UFL can beapplied
20 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Demand Reduction
Implementation
Solving LP for (x∗, y∗)
(x, y) = (x∗, y∗) round down to integer
(x, y) = (x∗, y∗)− (x, y), fractional part
rj =∑
i xij for I, rj = rj − rj for I(x, y) (integral) feasible and optimal for I(x, y) (fractional) feasible and optimal for I
Properties
rj = poly(|F |)ρ-approx for I implies ρ-approx for I
21 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Demand Reduction: Consequences
FTFP to FTFL, 1.7245-approximation
Sites into facilities
Clients with demand rj
FTFL size polynomial because of demand reduction
Ratio 1 + O(|F |/Q) for Q = minj rj , approaches 1 when Q is large
Next slide
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Ratio 1 + O(|F |/Q) for FTFP
LP x∗, y∗
x , yrj =
∑i xij
x , yrj =
∑i xij
x , yx + xy + y
LP-solver
round downto integer
fractional part
F ,C , fi , dij
F ,C , fi , dij
F ,C , fi , dij
UFL Instancec-approx
(1 + c |F |/Q)-approxsolution
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Ratio 1 + O(|F |/Q) for FTFP
LP x∗, y∗
x , yrj =
∑i xij
x , yrj =
∑i xij
x , yx + xy + y
LP-solver
round downto integer
fractional part
F ,C , fi , dij
F ,C , fi , dij
F ,C , fi , dij
UFL Instancec-approx
(1 + c |F |/Q)-approxsolution
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Techniques
Demand Reduction
Reduce all rj to polynomial values (to ensurepolynomial time of rounding)
ρ-approx for reduced instance ⇒ ρ-approx for originalinstance
Adaptive Partitioning
Split sites into facilities and clients into unit demands
Split associated fractional values
Properties ensure rounding similar to UFL can beapplied
24 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Adaptive Partitioning
site
i
client
j
y∗ix∗ij
facility
µ∑yµ = y∗i
unit demand
νxµν
xµν ∈ yµ, 0completeness
∑xµν = x∗ij
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Adaptive Partitioning
site
i
client
j
y∗ix∗ij
facility
µ∑yµ = y∗i
unit demand
νxµν
xµν ∈ yµ, 0completeness
∑xµν = x∗ij
25 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Adaptive Partitioning
site
i
client
j
y∗ix∗ij
facility
µ∑yµ = y∗i
unit demand
ν
xµν
xµν ∈ yµ, 0completeness
∑xµν = x∗ij
25 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Adaptive Partitioning
site
i
client
j
y∗ix∗ij
facility
µ∑yµ = y∗i
unit demand
νxµν
xµν ∈ yµ, 0completeness
∑xµν = x∗ij
25 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Neighborhood of Demand
xµν = yµ > 0
x = 0
neighborhood N(ν)
non-neighbor
demand ν∑µ xµν = 1
ν needs 1 facility...
Strategy 1: for each ν, open oneµ ∈ N(ν) with prob. yµ
optimal connection cost
large facility cost
Strategy 2: open facility only fordemands with disjoint
neighborhoods
optimal facility cost
large connection costHow to balance these two costs?
26 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Neighborhood of Demand
xµν = yµ > 0
x = 0
neighborhood N(ν)
non-neighbor
demand ν∑µ xµν = 1
ν needs 1 facility...
Strategy 1: for each ν, open oneµ ∈ N(ν) with prob. yµ
optimal connection cost
large facility cost
Strategy 2: open facility only fordemands with disjoint
neighborhoods
optimal facility cost
large connection costHow to balance these two costs?
26 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Neighborhood of Demand
xµν = yµ > 0
x = 0
neighborhood N(ν)
non-neighbor
demand ν∑µ xµν = 1
ν needs 1 facility...
Strategy 1: for each ν, open oneµ ∈ N(ν) with prob. yµ
optimal connection cost
large facility cost
Strategy 2: open facility only fordemands with disjoint
neighborhoods
optimal facility cost
large connection cost
How to balance these two costs?
26 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Neighborhood of Demand
xµν = yµ > 0
x = 0
neighborhood N(ν)
non-neighbor
demand ν∑µ xµν = 1
ν needs 1 facility...
Strategy 1: for each ν, open oneµ ∈ N(ν) with prob. yµ
optimal connection cost
large facility cost
Strategy 2: open facility only fordemands with disjoint
neighborhoods
optimal facility cost
large connection costHow to balance these two costs?26 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Two Types of Demands: Primary and Non-primary
κ1N(κ1)
primary
ν1non-primary
N(ν1)
assign
κ2
N(κ2)
N(κ1) ∩ N(κ2) = ∅
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Neighborhood Structure for Siblings
ν1
ν2
N(ν1)
N(ν2)
j
κ1
κ2
N(κ1)
N(κ2)
For siblings(N(κ1) ∪ N(ν1))
∩ (N(κ2) ∪ N(ν2)) = ∅
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Neighborhood Structure for Siblings
ν1
ν2
N(ν1)
N(ν2)
j
κ1
κ2
N(κ1)
N(κ2)
For siblings(N(κ1) ∪ N(ν1))
∩ (N(κ2) ∪ N(ν2)) = ∅
28 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Example of Partitioning
2/3
1/3
2/3
4/3
2/3
3
2
1
r3 = 1
r2 = 2
r1 = 2
Before Partitioning
2/3
1/3
2/3
1/31
2/3
ν31
ν21
ν22
ν11
ν12
After Partitioning
29 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Example of Partitioning
2/3
1/3
2/3
4/3
2/3
3
2
1
r3 = 1
r2 = 2
r1 = 2
Before Partitioning
2/3
1/3
2/3
1/31
2/3
ν31
ν21
ν22
ν11
ν12
After Partitioning
29 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Summary of Partitioning
Partitioning:
Clients → demands
Sites → facilities
(x∗, y∗) → (x , y)∑µ xµν = 1
xµν = yµ or 0
Structure:
If κ1, κ2 primary thenN(κ1) ∩ N(κ2) = ∅
small facility cost
Each non-primary ν assignedto κ with
N(κ) ∩ N(ν) 6= ∅priority(κ) ≤ priority (ν)
small connection cost of ν
(N(κ1) ∪ N(ν1)) ∩(N(κ2) ∪ N(ν2)) = ∅
fault-tolerance
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Summary of Partitioning
Partitioning:
Clients → demands
Sites → facilities
(x∗, y∗) → (x , y)∑µ xµν = 1
xµν = yµ or 0
Structure:
If κ1, κ2 primary thenN(κ1) ∩ N(κ2) = ∅
small facility cost
Each non-primary ν assignedto κ with
N(κ) ∩ N(ν) 6= ∅priority(κ) ≤ priority (ν)
small connection cost of ν
(N(κ1) ∪ N(ν1)) ∩(N(κ2) ∪ N(ν2)) = ∅
fault-tolerance
30 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Summary of Partitioning
Partitioning:
Clients → demands
Sites → facilities
(x∗, y∗) → (x , y)∑µ xµν = 1
xµν = yµ or 0
Structure:
If κ1, κ2 primary thenN(κ1) ∩ N(κ2) = ∅
small facility cost
Each non-primary ν assignedto κ with
N(κ) ∩ N(ν) 6= ∅priority(κ) ≤ priority (ν)
small connection cost of ν
(N(κ1) ∪ N(ν1)) ∩(N(κ2) ∪ N(ν2)) = ∅
fault-tolerance
30 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Summary of Partitioning
Partitioning:
Clients → demands
Sites → facilities
(x∗, y∗) → (x , y)∑µ xµν = 1
xµν = yµ or 0
Structure:
If κ1, κ2 primary thenN(κ1) ∩ N(κ2) = ∅
small facility cost
Each non-primary ν assignedto κ with
N(κ) ∩ N(ν) 6= ∅priority(κ) ≤ priority (ν)
small connection cost of ν
(N(κ1) ∪ N(ν1)) ∩(N(κ2) ∪ N(ν2)) = ∅
fault-tolerance
30 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Partitioning Implementation
Partitioning implementation: two phases
Phase 1, the partitioning phase
Define demands
Allocate facilities
Phase 2, the augmenting phase
Add facilities to make neighborhood unit
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 1, Step 1: Choose Best Client
In each iteration, create one demand for best client
j N(j)
N1(j): nearest unit chunk
bid(j) = avgdist(N1(j)) + α∗j (dual value)
Best bid client p selected to create a demand
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 1, Step 1: Choose Best Client
In each iteration, create one demand for best client
j N(j)
N1(j): nearest unit chunk
bid(j) = avgdist(N1(j)) + α∗j (dual value)
Best bid client p selected to create a demand
32 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 1, Step 2: Decide Neighborhood
Best client p creates demand ν, to decide N(ν), two cases:
N(κ1)
N(κ2)
. . .
N(κt)
N1(p)
disjoint
Case 1
N(κ) N1(p)
N1(p) overlaps some N(κ)
Case 2
33 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 1, Step 2 Contd.
Best client p creates demand ν, to decide N(ν), two cases:
Case 1: disjoint, N(ν) gets N1(p)
p1
2
34
N1(p)
N(p)
p3
4
ν1
2
Case 2: overlap, N(ν) gets N(p) ∩ N(κ)
p
12
3
4
5
67
N(κ)
N(p)N1(p)
p
15 6
7
ν2
34
34 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 1, Step 2 Contd.
Best client p creates demand ν, to decide N(ν), two cases:
Case 1: disjoint, N(ν) gets N1(p)
p1
2
34
N1(p)
N(p)
p3
4
ν1
2
Case 2: overlap, N(ν) gets N(p) ∩ N(κ)
p
12
3
4
5
67
N(κ)
N(p)N1(p)
p
15 6
7
ν2
34
34 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 1, Step 2 Contd.
Best client p creates demand ν, to decide N(ν), two cases:
Case 1: disjoint, N(ν) gets N1(p)
p1
2
34
N1(p)
N(p)
p3
4
ν1
2
Case 2: overlap, N(ν) gets N(p) ∩ N(κ)
p
12
3
4
5
67
N(κ)
N(p)N1(p)
p
15 6
7
ν2
34
34 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 2
Add facilities from N(j) to N(ν) until total value 1
N(j)
j
N(ν)
ν
1
2
3
4
5
6
7
2
57
35 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 2
Add facilities from N(j) to N(ν) until total value 1
N(j)
j
N(ν)
ν
1
2
3
4
5
6
72
57
35 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 2
Add facilities from N(j) to N(ν) until total value 1
N(j)
j
N(ν)
ν
1
2
3
4
5
6
72
5
7
35 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
Phase 2
Add facilities from N(j) to N(ν) until total value 1
N(j)
j
N(ν)
ν
1
2
3
4
5
6
72
57
35 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Demand Reduction Adaptive Partitioning
LP x∗, y∗
x , yrj =
∑i xij
x , yrj =
∑i xij
x , yx , yx + xy + y
ρ-approximatesolution
LP-solver
round downto integer
fractional part
adaptivepartitioning
LP-rounding
Done with partitioning, next to rounding36 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Rounding Dual-fitting
Outline
1 The FTFP Problem
2 Results in Dissertation
3 Related Work
4 Techniques
5 Approximation Algorithms
6 Summary
37 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Rounding Dual-fitting
3-Approximation for FTFP
Rounding: round each yµ and xµν to 0 or 1
Facilities: each primary κ opens one µ ∈ N(κ)
Connections: non-primary demands ν assigned to κconnect to µ
Analysis
Fault-Tolerance: ν uses only facilities in N(ν) ∪ N(κ)Cost: ≤ 3 · LP∗, because
Facility cost ≤ F ∗
Connection cost ≤ C∗ + 2 · LP∗
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FTFP Results Related Work Techniques Algorithms End Rounding Dual-fitting
1.736-Approximation for FTFP
Rounding: round each yµ and xµν to 0 or 1
Facilities:Each primary κ opens random µ ∈ N(κ)Other facilities open randomly independently
Connections:If a neighbor open, connect to nearest neighborElse connect via assigned primary demand
Analysis
Fault-Tolerance: ν uses only facilities in N(ν) ∪ N(κ)Cost: ≤ (1 + 2/e)LP∗, because
Facility cost ≤ F ∗
Connection cost ≤ C∗ + (2/e) · LP∗
39 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Rounding Dual-fitting
1.575-Approximation for FTFP — Partitioning
More intricate neighborhood structure
Two neighborhoods: close and far, N(ν) = Ncls(ν) ∪ Nfar(ν)
Ncls(ν) = nearest (1/γ)-fraction of N(ν)
Ncls(ν) ∩ Ncls(κ) 6= ∅, if ν assigned to κ
For siblings ν1, ν2, Ncls(κ1) ∪ N(ν1) and Ncls(κ2) ∪ N(ν2)disjoint
...
ν
Ncls(ν)
Nfar(ν)
40 / 45 [email protected] Approximation Algorithms for FTFP
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FTFP Results Related Work Techniques Algorithms End Rounding Dual-fitting
1.575-Approximation for FTFP — Rounding
Rounding: boost (x∗, y∗) by γ and apply demand reductionand adaptive partitioning, then round by
Facilities:Each primary κ opens random µ ∈ Ncls(κ)Other facilities open randomly independently
Connections:If a neighbor open, connect to nearest neighborElse connect via assigned primary demand
Analysis
Fault-Tolerance: ν uses only facilities in N(ν) ∪ Ncls(κ)Cost: ≤ γ · LP for γ = 1.575, because
Facility cost ≤ γ · F ∗
Connection cost ≤ γ · C∗
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FTFP Results Related Work Techniques Algorithms End Rounding Dual-fitting
Greedy and Dual-fitting
Greedy in polynomial time
Best star can be found quicklyBest star remains best
Ratio Hn (Wolsey’s result): Greedyis Hn-approx for
Minimizing a linear functionSubject to submodularconstraints
Lower bound Ω(log n/ log log n) fordual-fitting
Example has k groups, n = kk
Shrinking factor is k/2
i
i
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FTFP Results Related Work Techniques Algorithms End Rounding Dual-fitting
Dual-fitting Example
Dual constraints force a ratio of k/2, number of clients n = kk
f1 n1 = kk , r1
n2 = kk−1, r2
n3 = kk−2, r3
nk = k , rk
demands r1 r2 . . . rk
d1 = 0
d2 = d1 + f1/n1
d3 = d2 + f1/n2
dk = dk−1 + f1/nk−1
first r1 stars (f1, n1)
next r2 stars (f1, n2)
next r3 stars (f1, n3)
next r1 stars (f1, n1)
next rk stars (f1, nk)
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FTFP Results Related Work Techniques Algorithms End
Outline
1 The FTFP Problem
2 Results in Dissertation
3 Related Work
4 Techniques
5 Approximation Algorithms
6 Summary
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FTFP Results Related Work Techniques Algorithms End
Summary
Results
1.575-approximation algorithm for FTFP
Technique for extending LP-rounding algorithms for UFLto FTFP
Open Problems
Can FTFL be approximated with the same ratio?
LP-free algorithms for FTFP or FTFL with constant ratio?
Close the 1.463− 1.488 gap for UFL!
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FTFP Results Related Work Techniques Algorithms End
Summary
Results
1.575-approximation algorithm for FTFP
Technique for extending LP-rounding algorithms for UFLto FTFP
Open Problems
Can FTFL be approximated with the same ratio?
LP-free algorithms for FTFP or FTFL with constant ratio?
Close the 1.463− 1.488 gap for UFL!
45 / 45 [email protected] Approximation Algorithms for FTFP