Approximation Algorithms for Single and Multi-Commodity Connected Facility Location. Fabrizio Grandoni & Thomas Rothvoß
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1 Approximation Algorithms for Single and Multi-Commodity Connected Facility Location Fabrizio Grandoni & Thomas Rothvoß Department of Mathematics, M.I.T. IPCO 2011
2 Outline 1. Better approximation algorithm for Connected Facility Location
3 Outline Better approximation algorithm for Connected Facility Location First O(1)-apx for Multi-Commodity Connected Facility Location
4 Outline Better approximation algorithm for Connected Facility Location First O(1)-apx for Multi-Commodity Connected Facility Location 3. Improved hardness results for several problems
5 Part 1: Connected Facility Location Given: graph G = (V,E), with metric distances c : E Q +
6 Part 1: Connected Facility Location Given: graph G = (V,E), with metric distances c : E Q + a set of clients C V clients
7 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
8 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
9 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 Goal: open facilities F F facility clients minimizing f F o(f) opening cost + j Cc(j,F ) connection cost
10 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 Goal: open facilities F F facility find Steiner tree T spanning opened facilities minimizing f F o(f) opening cost + j Cc(j,F ) connection cost + M e T c(e) Steiner cost clients
11 Previous Results for Connected Facility Location APX-hard (reduction from Steiner Tree) apx based on LP-rounding [Gupta et al 01] apx primal-dual algorithm [Swamy, Kumar 02] apx based on random-sampling [Eisenbrand, Grandoni, R., Schäfer 08]
12 Previous Results for Connected Facility Location APX-hard (reduction from Steiner Tree) apx based on LP-rounding [Gupta et al 01] apx primal-dual algorithm [Swamy, Kumar 02] apx based on random-sampling [Eisenbrand, Grandoni, R., Schäfer 08] Our result Simple 3.19-approximation algorithm.
13 Previous Results for Connected Facility Location APX-hard (reduction from Steiner Tree) apx based on LP-rounding [Gupta et al 01] apx primal-dual algorithm [Swamy, Kumar 02] 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]
14 Where is the difficulty? Algorithm: opening cost 0
15 Where is the difficulty? Algorithm: (1) Just solve Facility Location opening cost 0
16 Where is the difficulty? Algorithm: (1) Just solve Facility Location (2) Run Steiner Tree algorithm opening cost 0 pay tree M times!
17 Where is the difficulty? Algorithm: (1) Just solve Facility Location (2) Run Steiner Tree algorithm pay tree M times! Doesn t work! opening cost 0
18 The CFL algorithm
19 The CFL algorithm (1) Guess facility r from OPT. r
20 The CFL algorithm (1) Guess facility r from OPT. (2) Sample each client with prob 1 M C r
21 The CFL algorithm (1) Guess facility r from OPT. (2) Sample each client with prob 1 M C (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
22 The CFL algorithm (1) Guess facility r from OPT. (2) Sample each client with prob 1 M C (3) Compute 1.5-apx F for Facility Location instance o (f) := o(f)+m c(f,nearest node in C {r}) r
23 The CFL algorithm (1) Guess facility r from OPT. (2) Sample each client with prob 1 M C (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
24 The CFL algorithm (1) Guess facility r from OPT. (2) Sample each client with prob 1 M C (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: E[APX] E cost of apx Facility Location sol. +penalties +E cost of apx Steiner Tree
25 The CFL algorithm (1) Guess facility r from OPT. (2) Sample each client with prob 1 M C (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: E[APX] E cost of apx Facility Location sol. +penalties +E cost of apx Steiner Tree on C {r}
26 Analysis: Steiner Cost Notation: OPT = O opening cost + S Steiner tree cost + C connection cost
27 Analysis: Steiner Cost Notation: Lemma OPT = O opening cost + S Steiner tree cost + C connection cost E[M apx Steiner Tree on C {r}] 1.39 (S +C ) r
28 Analysis: Steiner Cost Notation: Lemma OPT = O opening cost + S Steiner tree cost + C connection cost E[M apx Steiner Tree on C {r}] 1.39 (S +C ) r S Steiner tree in OPT
29 Analysis: Steiner Cost Notation: Lemma OPT = O opening cost + S Steiner tree cost + C connection cost E[M apx Steiner Tree on C {r}] 1.39 (S +C ) r S Steiner tree in OPT + 1 M C } M {{} need path in C with prob. 1/M
30 Analysis: Steiner Cost Notation: Lemma OPT = O opening cost + S Steiner tree cost + C connection cost E[M apx Steiner Tree on C {r}] 1.39 (S +C ) r ( 1.39 S Steiner tree in OPT + 1 ) M C } M {{} need path in C with prob. 1/M
31 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 v D ] dist(v, sampled node {r}) 0.81 p c(t) r T
32 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 v D ] dist(v, sampled node {r}) 0.81 p c(t) r T
33 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 v D ] dist(v, sampled node {r}) 0.81 p c(t) r T
34 Analysis: Facility Location cost Theorem E[apx Facility Location cost] 1.5 (O +2C +0.81S ) r
35 Analysis: Facility Location cost Theorem E[apx Facility Location cost] 1.5 (O +2C +0.81S ) There is a good sol: F := facilities in OPT serving C {r} r
36 Analysis: Facility Location cost Theorem E[apx Facility Location cost] 1.5 (O +2C +0.81S ) There [ is a good sol: ] F := facilities in OPT serving C {r} opening cost E O + penalties + 1 M C M orig. cost expected penalty r
37 Analysis: Facility Location cost Theorem E[apx Facility Location cost] 1.5 (O +2C +0.81S ) There [ is a good sol: ] F := facilities in OPT serving C {r} opening cost E O + penalties + 1 M C M orig. cost expected penalty E[connection cost] C + r
38 Analysis: Facility Location cost Theorem E[apx Facility Location cost] 1.5 (O +2C +0.81S ) There [ is a good sol: ] F := facilities in OPT serving C {r} opening cost E O + penalties + 1 M C M orig. cost expected penalty E[connection cost] C /M S M tree Use Core Detouring Theorem with T = S and p := 1 M. r
39 Finishing the analysis for CFL Conclusion: E[APX] 1.39 (S +C )+1.5 (O +2C +0.81S )
40 Finishing the analysis for CFL Conclusion: E[APX] 1.39 (S +C )+1.5 (O +2C +0.81S ) 1.5 O +2.7 S C
41 Finishing the analysis for CFL Conclusion: E[APX] 1.39 (S +C )+1.5 (O +2C +0.81S ) 1.5 O +2.7 S C Improvement to 3.19: Adapting the sampling probability Using a bi-factor Facility Location algorithm
42 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 s 6 s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 r 6 s 2 r 3
43 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 Goal: Find Facilities F F f F o(f) opening cost s 6 s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 r 6 s 2 r 3
44 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 Goal: Find Facilities F F Forest T f F o(f) opening cost s 6 +M c(t) forest cost s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 r 6 s 2 r 3
45 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 Goal: Find Facilities F F Forest T f F o(f) opening cost s 6 +M c(t) + forest cost s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 k dist F,T(s i,r i ) 0 distance between opened & connected facilities i=1 r 6 s 2 r 3
46 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 Goal: Find Facilities F F Forest T f F o(f) opening cost s 6 +M c(t) + forest cost s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 k dist F,T(s i,r i ) 0 distance between opened & connected facilities i=1 r 6 s 2 r 3
47 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 Goal: Find Facilities F F Forest T f F o(f) opening cost s 6 +M c(t) + forest cost s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 k dist F,T(s i,r i ) 0 distance between opened & connected facilities i=1 r 6 s 2 r 3
48 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 Goal: Find Facilities F F Forest T f F o(f) opening cost s 6 +M c(t) + forest cost s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 k dist F,T(s i,r i ) 0 distance between opened & connected facilities i=1 r 6 s 2 r 3
49 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 Goal: Find Facilities F F Forest T f F o(f) opening cost s 6 +M c(t) + forest cost s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 k dist F,T(s i,r i ) 0 distance between opened & connected facilities i=1 r 6 s 2 r 3
50 Part 2: Multi-Commodity Connected Facility Location Input: Undirected graph G = (V,E), metric distances c : E Q + source-sink pairs (s 1,r 1 ),...,(s k,r k ) a set of facilities F V with opening costs o : F Q + parameter M 1 Goal: Find Facilities F F Forest T f F o(f) opening cost s 6 +M c(t) + forest cost s 5 r 5 s 3 s 1 r 1 s 4 r 2 r 4 k dist F,T(s i,r i ) 0 distance between opened & connected facilities i=1 r 6 s 2 r 3
51 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
52 Part 3: Improved hardness results Steiner Tree Facility Location Single-Sink Rent-or-Buy generalizes Connected Facility Location Single-Sink Buy-at-Bulk Virtual Private Network
53 Part 3: Improved hardness results Steiner Tree 1.01 Facility Location 1.46 Single-Sink Rent-or-Buy 1.01 generalizes Connected Facility Location 1.46 Single-Sink Buy-at-Bulk 1.01 Virtual Private Network 1.01
54 Part 3: Improved hardness results Steiner Tree 1.01 Facility Location 1.46 Single-Sink Rent-or-Buy generalizes Connected Facility Location 1.46 Single-Sink Buy-at-Bulk Virtual Private Network
55 Part 3: Improved hardness results Steiner Tree 1.01 Facility Location 1.46 o(f) = 0 f V Single-Sink Rent-or-Buy generalizes Connected Facility Location 1.46 Single-Sink Buy-at-Bulk Virtual Private Network
56 The reduction (1) Reduce Set Cover to SROB elements sets
57 The reduction (1) Reduce Set Cover to SROB clients/elements facilities/sets c(e) {1,2} M = 0.27n OPT SC r
58 The reduction (1) Reduce Set Cover to SROB clients/elements facilities/sets 1 2 c(e) {1,2} M = 0.27n OPT SC r
59 The reduction (1) Reduce Set Cover to SROB remove! clients/elements facilities/sets 1 2 buy! c(e) {1,2} M = 0.27n OPT SC r (1) WHILE not all elements covered DO (2) Compute 1.27-apx SROB sol (3) Buy facilities/sets in sol. & remove covered elements
60 The reduction (2) Use idea from [Guha & Khuller 99] # clients at dist 1 n 0.27 # fac OPT SC n n 1 OPT SC 2 OPT SC # facilities in apx. sol
61 The reduction (2) Use idea from [Guha & Khuller 99] # clients at dist 1 n apx SROB sol # fac OPT SC n n 1 OPT SC 2 OPT SC # facilities in apx. sol
62 The reduction (2) Use idea from [Guha & Khuller 99] # clients at dist 1 n apx SROB sol # fac OPT SC n n 1 OPT SC 2 OPT SC # facilities in apx. sol # needed sets [...some calc...] 0.999ln(n) OPT SC
63 The reduction (2) Use idea from [Guha & Khuller 99] # clients at dist 1 n apx SROB sol # fac OPT SC n n 1 OPT SC 2 OPT SC # facilities in apx. sol # needed sets [...some calc...] 0.999ln(n) OPT SC Contradiction! Theorem (Feige 98) Unless NP DTIME(n O(loglogn) ), there is no (1 ε) ln(n)-apx for Set Cover.
64 The end Thanks for your attention
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