Download presentation

Presentation is loading. Please wait.

Published byAlejandro Colie Modified over 4 years ago

1
A Dependent LP-Rounding Approach for the k-Median Problem Moses Charikar 1 Shi Li 1 1 Department of Computer Science Princeton University ICALP 2012, Warwick, UK

2
Introduction Linear Programming Relaxation Simple Pseudo-Approx. for k-median Our Algorithm for k-medianOutline

3
k-Median as a Clustering Problem Given: metric (X, d), k Partition X into k clusters Select a center for each cluster Minimize sum of distances to the centers: Quantifies how well a set can be divided into k partitions k = 4

4
k-Median in Operation Research Given metric (F C, d), k o F : set of facilities o C : set of clients Open k facilities Connect each client to its nearest open facility Minimize total connection cost k = 4

5
Related Problem : Facility Location Problem Given metric (F C, d), k o F : set of facilities o C : set of clients o f i : facility cost of opening i Open k facilities Connect each client to its nearest open facility Minimize total connection cost {f i ≥ 0 : i F} Open a set F' F of facilities Minimize sum of facility cost and connection cost, k = 4

6
Known Results * local search: if switching p facilities can not improve a solution, then the solution is a 3+2/p-approx. Integrality gap of the natural linear programming is between 2 and 3 o the proof of the upper bound 3 is non-constructive Approx.Hardness of appox. facility location1.488 [Li11]1.463 [GK98,Sri02] k-median3+ε * [AGK + 01]1+2/e+ε [JMS02]

7
Our Results A LP-rounding approach for k-median o prove 3.25 approximation ratio o thus give a constructive proof for the 3.25 integrality gap o faster running time compared to the local search algorithm o potential to improve the 3+ε approximation the upper bound 3.25 is not tight our algorithm may already give approximation ratio smaller than 3

8
Our Results prev. best approx. ratioour approx. ratio k-facility location [Zha06]3.25 matroid median16 [KKN + 11]9 knapsack median ≥ 1000 [Kum12]34 k-facility location: facility location problem with constraint that at most k facilities can be open matroid median: the set of open facilities must be an independent set of a given matroid knapsack median problem: each facility has a cost, the total cost of open facilities can not exceed a budget B

9
Introduction Linear Programming Relaxation Simple Pseudo-Approx. for k-median Our Algorithm for k-medianOutline

10
Natural LP Relaxation y i {0,1}, i F : whether facility i is open x i,j {0,1}, i F, j C : whether client i is connected to facility j Every client j must be connected to 1 facility Client j can only be connected to an open facility We can open at most k facilities

11
Canonical Instance km facilities every client j is connected to its nearest m facilities in the LP solution, y i =1/m, x i,j {0,1/m} facilities clients j

12
Canonical Instance F j : the set of m facilities that j is connected to average distance from j to F j maximum distance from j to F j LP value = facilities clients j

13
Introduction Linear Programming Relaxation Simple Pseudo-Approx. for k-median Our Algorithm for k-medianOutline

14
Pseudo-Approximation An (α, c)-pseudo approximation is a solution that opens at most αk facilities and whose connection cost is at most c times the optimal cost A warm-up : (1 + ε, O(1/ε))-pseudo approximation for k-median

15
Pseudo-Approximation Let m' = m / (1+ε), y' i =(1+ε)y i =1/m' Every client only needs to connect to m' facilities We fractionally open km(1/m')=(1+ε)k facilities Define F' j, d' av (j),d' max (j) similarly facilities clients j

16
Pseudo-Approximation Two clients j and j' conflict if F' j F' j' ≠ ∅ Select a set C' of clients such that no two clients in C' conflict each other facilities clients j j'

17
Pseudo-Approximation greedily constructing C' C with no confliction o while C ≠ ∅, select j C with the minimum d av (j) add j to C' remove j and all clients that conflict j from C facilities clients

18
Pseudo-Approximation open facilities o For every j C', randomly open 1 of the m' facility in F' j o For any facility i that is not inside j C' F' j, open i with probability 1/m' connect each client to its nearest open facility facilities clients Fact: every facility is open with probability 1/m'

19
Pseudo-Approximation j j' facilities clients F' j F' j'

20
Introduction Linear Programming Relaxation Simple Pseudo-Approx. for k-median Our Algorithm for k-medianOutline

21
Barrier to Obtain True Approximation If ε=0, then F' j =F j d max (j) >> d av (j) With non-zero prob., j will be connected to facilities in F j' The expected connection cost of j is unbounded compared to d av (j) facilities clients FjFj F j' j j'

22
Remove the Barrier Solution: j only “claims” close facilities in F j Let U j be the set of claimed facilities Use U j to replace F j in the algorithm New Barrier: |U j | < m might happen can not guarantee always a facility open in U j FjFj UjUj j

23
Remove the New Barrier can guarantee |U j | ≥ m/2 |U j U j' | ≥ m if U j and U j' are disjoint pair the clients in C' always open 1 facility (possibly 2 facilities) in U j U j' for a matched pair (j, j') j UjUj U j' j'

24
Remove the New Barrier How to open facilities for a matched pair? m boxes in a line Permute facilities in U j put them in the leftmost |U j | boxes Permute facilities in U j' put them in the rightmost |U j' | boxes Open facilities in a random selected box m UjUj U j'

25
The Algorithm Filtering o 2 clients j and j' conflict if d(j, j') ≤ 4max{d av (j),d av (j')} o while C ≠ ∅ select j C that minimizes d av (j); add j to C' remove j and all clients that conflict j from C

26
The Algorithm Filtering Claiming o For any j C', let 2R j be the distance between j and its nearest neighbor in C' o A facility i is claimed by j, if i F j and d(i, j) ≤ R j i.e, U j = F j Ball(j, R j ) Fact: any client j C' will claim at least m/2 and at most m facilities.

27
The Algorithm Filtering Claiming Matching o while there are at least 2 unmatched clients in C' select 2 unmatched clients j and j' that minimizes d(j, j') match j and j'

28
The Algorithm Filtering Claiming Matching Rounding o For each matched pair (j, j'), open 1 or 2 facilities in U j U j' o If there is an unmatched client j, open 0 or 1 facility in U j o For each facility i that is not inside any U j, open i with probability 1/m o Connect each client to its nearest open facility

29
Proof of Constant Approx. Ratio

30
j j1j1 j2j2 nearest neighbor of j in C' j 2 is matched with j 1 2Rj2Rj 2R j1 ≤ 2R j RjRj Rj1Rj1 Rj2Rj2 UjUj Uj1Uj1 Uj2Uj2

31
Proof of 3.25 approx. ratio complicated, details omitted rough idea : for a client j C' o j 1 C' is the client that conflicts and removes j in the filtering phase o j 2 C' is the nearest neighbor of j 1 in C' o j 3 C' is the client matched with j 2 o Consider the nearest open facility of j in F j F j1 U j2 U j3 Our algorithm opens k facilities in expectation Can be easily transformed so that it always opens k facilities Algorithm naturally extends to k-FL problem

32
Ongoing Work Joint work with Svensson, improved the best approximation ratio (3+ε) for k-median

33
Summary We introduced a LP-rounding algorithm for k-median problem o proved 3.25 approximation ratio for the problem o it has potential to improve the decade-long 3 approximation Improved approximation algorithms for the following problems o k-facility location problem 3.25 o Matroid median problem 9 o Knapsack median problem 34

Similar presentations

OK

1 On Completing Latin Squares Iman Hajirasouliha Joint work with Hossein Jowhari, Ravi Kumar, and Ravi Sundaram.

1 On Completing Latin Squares Iman Hajirasouliha Joint work with Hossein Jowhari, Ravi Kumar, and Ravi Sundaram.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google