Download presentation

Presentation is loading. Please wait.

Published byWarren Foskey Modified over 2 years ago

1
Primal-Dual Algorithms for Connected Facility Location Chaitanya SwamyAmit Kumar Cornell University

2
Theory Seminar, 10/2002 Connected Facility Location (ConFL) F : set of facilities. D : set of clients(demands). Facility i has facility cost f i. c ij : distance between i and j in V. client facility Graph G=(V,E), costs {c e } on edges and a parameter M ≥ 1. node

3
Theory Seminar, 10/2002 We want to : client facility Cost = i A f i + j D c i(j)j + M e T c e = facility opening cost + client assignment cost + cost of connecting facilities. 2)Assign each demand j to an open facility i(j). Steiner tree 3)Connect all open facilities by a Steiner tree T. open facility 1)Pick a set A of facilities to open.

4
Theory Seminar, 10/2002 Ravi & Selman : Look at related problem - connect facilities by a tour. Round optimal solution of an exponential size LP. Karger & Minkoff : Combinatorial algorithm with ‘large’ constant approx. ratio. Gupta, Kleinberg, Kumar, Rastogi & Yener : Also use LP rounding. Get a 9.001-approx. when F =V, f i = 0 for all i, and a 10.66-approx. in the general case. Previous Work

5
Theory Seminar, 10/2002 Our Results Give primal-dual algorithms with approx. ratios of 5 ( F =V, f i = 0) and 9 (general case). Use this to solve the Connected k-Median problem, and an edge capacitated version of ConFL.

6
Theory Seminar, 10/2002 Rent-or-buy problem Special case with F =V and f i = 0, i. Suppose we know a facility f that is open in an optimal solution. What does the solution look like ? Steiner tree. : cost of e = Mc e. Shortest paths : cost of e = c e (demand through e). f Open facility

7
Theory Seminar, 10/2002 Equivalently, Want to route traffic from clients to f by installing capacity on edges. 2 choices : I : Rent Capacity cost rented capacity (Shortest Paths) II : Buy Capacity fixed cost, ∞ capacity (Steiner Tree) M cost capacity ConFL single-sink buy-at-bulk problem with 2 cable types and sink f.

8
Theory Seminar, 10/2002 A Naive Algorithm 1)Run a -approx. algorithm for FL. 2)Build a Steiner tree on open facilities. This will just open a facility at each demand. Can be as bad as (M) times OPT. e.g., n 1 11 1 n Naive algorithm OPT Cost = 2M(n+1). Cost = 2M+2n. client node open facility f f

9
Theory Seminar, 10/2002 Need to cluster enough demand at a facility before opening it. Suppose in naive algorithm, each open facility serves ≥ M clients. Total cost of adding edges ≤ C * +C. Opt. Steiner tree on ≤ S * +C * +C. S ≤ 2(S * +C * +C) get an O(1)-approx. ≥ M C S*S* C*C* OPT C *, S * : assignment, Steiner cost of OPT.

10
Theory Seminar, 10/2002 An Integer Program x ij = 1 if demand j is assigned to i. z e = 1 if edge e is included in the Steiner tree. Min. j,i c ij x ij + M e c e z e (primal) s.t. i x ij ≥ 1 j i S x ij ≤ e (S) z e S V : f S, j x ij, z e {0, 1}(1) Relax (1) to x ij, z e ≥ 0 to get an LP. f S j

11
Theory Seminar, 10/2002 What is the dual? v j amount j is willing to ‘pay’ to route its demand to f. Let y S = j y S,j. {y S } moat packing around facilities. v j ≤ c ij + S:i S,f S y S,j Any feasible dual solution is a lower bound on OPT – Weak Duality. Max. j v j c ij j f S:e (S),f S y S ≤ Mc e

12
Theory Seminar, 10/2002 The Primal-Dual method Bar-Yehuda & Even : First primal-dual approx. algorithm for vertex cover. Agrawal, Klein & Ravi : Gave a more sophisticated algorithm for the generalized Steiner problem. Goemans & Williamson : Extended the schema to a large class of network design problems. Jain & Vazirani : Used a different approach to solve the facility location problem and its variants. 1)Construct primal soln. and dual soln. simultaneously. 2)Bound primal cost by c (dual soln.) get a c-approx. algorithm.

13
Theory Seminar, 10/2002 A Primal-Dual algorithm 1)Decide which facilities to open, cluster demands around open facilities. Construct a primal soln. and dual soln. (v,y) simultaneously. 2)Build a Steiner tree on open facilities. The following will hold after 1) : (v,y) is a feasible dual solution. j is assigned to i(j) s.t. c i(j)j ≤ 3v j. Each open facility i has ≥ M clients, {j}, assigned to it s.t. c ij ≤ v j.

14
Theory Seminar, 10/2002 Analysis Suppose the 3 properties hold. Let A be the set of facilities opened, D i = {j assigned to i : c ij ≤ v j } for i A. Then, |D i | ≥ M. Let D 1 = U i A D i. can amortize the cost against demands in D i. ≥ M DiDi So, S ≤ 2(S * + C * + j D 1 v j ). i C ≤ j D 1 v j + j D 1 3v j.

15
Theory Seminar, 10/2002 Need to cluster enough demand at a facility before opening it. Suppose in naive algorithm, each open facility serves ≥ M clients. Total cost of adding edges ≤ C * +C. Opt. Steiner tree on ≤ S * +C * +C. S ≤ 2(S * +C * +C) get an O(1)-approx. ≥ M C S*S* C*C* OPT C *, S * : assignment, Steiner cost of OPT.

16
Theory Seminar, 10/2002 Analysis Suppose the 3 properties hold. Let A be the set of facilities opened, D i = {j assigned to i : c ij ≤ v j } for i A. Then, |D i | ≥ M. Let D 1 = U i A D i. Total cost ≤ 2(S * + C * ) + 3 j D v j ≤ 5 OPT. can amortize the cost against demands in D i. ≥ M DiDi So, S ≤ 2(S * + C * + j D 1 v j ). i C ≤ j D 1 v j + j D 1 3v j.

17
Theory Seminar, 10/2002 Phase 1 Simplifying assumption – can open a facility anywhere along an edge. locations Notion of time, t. Start at t=0. Initially v j = 0, j. f is tentatively open, all other locations are closed. Say j is tight with i (has reached i) if v j ≥ c ij. S j is set of vertices j is tight with.

18
Theory Seminar, 10/2002 Keep raising v j s until : 1)There is a closed location i with which M demands become tight : tentatively open i. Freeze all these M demands. 2)j reaches a tentatively open location : freeze j. Now only raise v j of unfrozen demands. Continue this process until all demands are frozen. Raise all v j at rate 1. Also raise y S j,j at same rate.

19
Theory Seminar, 10/2002 Execution of the algorithm not open location tentatively open location unfrozen demand M=2, time t=0 f M=2, time t=1

20
Theory Seminar, 10/2002 Execution of the algorithm M=2, time t=2 f not open location tentatively open location unfrozen demand frozen demand

21
Theory Seminar, 10/2002 Execution of the algorithm M=2, time t=3 f frozen demand not open location tentatively open location unfrozen demand j v j =t=3

22
Theory Seminar, 10/2002 Opening locations Let A’= set of tentatively opened i. D i = {j : j is tight with i } for i A’. t i = time when i was tent’vely opened. Note, |D i | ≥ M. But the D i s may not be disjoint. 4 2 3 1 0 f Say i, l in A’ are dependent if D i D l . Consider locations in order of t i. If A U {i} is independent add i to A. Here order is 0,1,2,3,4 so A={0,1,3}. Initially A = Pick an indepdt. set of locations, A A’ and open these.

23
Theory Seminar, 10/2002 Assigning clients Consider client j. j k i l i is tentatively open and caused j to get frozen j is tight with i. i and l are dependent, t l ≤ t i and l is open : assign j to l. 2) open location tentatively open location, not open j is tight with some open i : assign j to i. At most 1 open i may be tight with j. 1) j i

24
Theory Seminar, 10/2002 i(j) = location that j is assigned to. 1)c i(j)j ≤ v j. 2)c i(j)j ≤ v j + 2v k. v j ≥ t i since j freezes at or after time t i. Also v k ≤ t i. So v k ≤ v j and c i(j)j ≤ 3v j. For every open i, all clients in D i are assigned to i. |D i | ≥ M, so at least M demands {j} are assigned to i with c ij ≤ v j.

25
Theory Seminar, 10/2002 Feasibility (v,y) : dual solution v j ≤ c ij + S:i S,f S y S,j i f, j Once j reaches i, set S j includes i and y S j,j is increasing at same rate as v j. j freezes when it reaches f. vjvj c ij j i S:i S,f S y S,j SjSj

26
Theory Seminar, 10/2002 j S:e (S),f S y S,j ≤ Mc e e S y S,j = portion of in j’s j ( S y S,j ) = p e (# of crossing p) M=3 e ≤ M at any location ≤ p e M = Mc e.

27
Theory Seminar, 10/2002 Done? Not quite. Could have opened non-vertex locations. Fix : Move all such locations to vertices. Cost only decreases ! e.g., ≥ M So, get a 5-approx. for rent-or-buy.

28
Theory Seminar, 10/2002 The General Case F V : Cannot open a facility everywhere. Facilities have costs : Want to open cheap facilities. Modified Phase 1 1)Dual variables, v j, also pay for opening facilities. 2)Cluster ≥ M demands around terminal locations, open facilities ‘near’ terminal locations. Theorem : There is a 9-approx. algorithm for the general case.

29
Theory Seminar, 10/2002 Let A be the set of terminal locations, D i = {j : c ij ≤ v j } for i A. We ensure |D i | ≥ M. Let D 1 = U i A D i. Total cost ≤ 2(S * + C * ) + 7 j D v j ≤ 9 OPT. ≥ M DiDi S ≤ j D 1 2v j + Can show, F + C ≤ j D 1 3v j + j D 1 7v j. ≤ 2 min j D i v j open facility 2(S * + C * + j D 1 v j ). i

30
Theory Seminar, 10/2002 Open Questions 1)Better approximation : only know an integrality gap of 2 from Steiner tree. 2)Multicommodity buy-at-bulk. Multiple source-sink pairs, route flow from source to sink. Have different cable types. Kumar et al. give a const. approx. for multicommodity rent-or-buy. 3)Unrelated metrics : Min. i f i y i + j,i c ij x ij + e d e z e. Can reduce from group Steiner tree – get a O(log 2 n) LB. Matching UB?

Similar presentations

OK

LP-Based Algorithms for Capacitated Facility Location Hyung-Chan An EPFL July 29, 2013 Joint work with Mohit Singh and Ola Svensson.

LP-Based Algorithms for Capacitated Facility Location Hyung-Chan An EPFL July 29, 2013 Joint work with Mohit Singh and Ola Svensson.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on culture of kerala Ppt on network theory business Ppt on cross-sectional study meaning Seminar ppt on cam and follower Ppt on road accidents statistics Ppt on holographic technology hologram Ppt on point contact diode Ppt on diode family video Ppt on basics of ms word 2007 Ppt on network security issues