1. Improved approximation for k-median Shi Li Department of Computer Science Princeton University Princeton, NJ, 08540 04/20/2013

2. $100 $130 maintenance cost transportation cost $10 $20 $50 $30 + minimize

3. BALINSKI, M. L.1966. On finding integer solutions to linear programs. In Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems. IBM, New York, pp. 225–248. KUEHN, A. A., AND HAMBURGER, M. J. 1963. A heuristic program for locating warehouses. STOLLSTEIMER, J. F.1961. The effect of technical change and output expansion on the optimum number, size and location of pear marketing facilities in a California pear producing region. Ph.D. thesis, Univ. California at Berkeley, Berkeley, Calif. STOLLSTEIMER, J. F.1963. A working model for plant numbers and locations. J. Farm Econom. 45, 631– 645. Facility Location Problem

4. Uncapacitated Facility Location (UFL) facility cost connection cost + F : potential facility locations C : set of clients f i, i  F : cost for opening i d : metric over F  C find S  F, minimize facilities clients $30 $100 $20 $100

5. Wal-mart Stores in New Jersey Question : Suppose you have budget for 50 stores, how will you select 50 locations?

6. k -median facilities clients + F : potential facility locations C : set of clients d : metric over F  C find S  F, minimize f i, i  F : cost for opening i k : number of facilities to open | S |= k

7. k -median clustering

8. Known Results: UFL O(log n)-approximation [Hoc82] constant approximations 3.16 [STA98] 2.41 [GK99] 3 [JV99] 1.853 [CG99] 1.728 [CG99] 5+ε [Kor00] 1.861 [MMSV01] 1.736 [CS03] 1.61 [JMS02] 1.582 [Svi02] 1.52 [MYZ02] 1.50 [Byr07] 1.488 [Li11] 1.463-hardness of approx. [GK98]

9. 4Deterministic rounding of linear programs 4.5 The uncapacitated facility location problem 5Random sampling and randomized rounding of linear programs 5.8 The uncapacitated facility location problem 7The primal-dual method 7.6 The uncapacitated facility location problem 9Further uses of greedy and local search algorithms 9.1 A local search algorithm for the uncapacitated facility location problem 9.4 A greedy algorithm for the uncapacitated facility location problem 12 Further uses of random sampling and randomized rounding of linear programmings 12.1 The uncapacitated facility location problem

10. Know results : k -median  pseudo-approximation  1-approx with O(k log n) facilities [Hoc82]  2(1+ε)-approx. with (1+1/ε)k facilities[LV92]  super-constant approximation  O(log n loglog n) [Bar96,Bar98]  O(log k loglog k) [CCGS98]

11. Known Results: k -median  constant approximation LP rounding Primal-Dual Local Search 6.667 [CGTS99]6 [JV99] 4 [CG99]4 [JMS03]3.25 [CL12] 3+ε [AGK + 01] 1+√3+ε [LS13]  (1+2/e)-hardness of approximation [JMS03]

12. Lloyd Algorithm[Lloyd82]  k-means clustering : min total squared distances  k-means vs k-median clustering: k-means is more often used Walmart example: k-median is more appropriate approximation: k-median is “easier”

13. Local Search  Can we improve the solution by p swaps?  No : stop  Yes : swap and repeat  Approximation :  k-median : 3+2/p [AGK + 01]  k-means : (3+2/p) 2 [KMN + 02]

14. LP for k -median y i : whether to open i x i,j : whether connect j to i open at most k facilities client j must be connected client j can only connected to an open facility integrality gap is at least 2 integrality gap is at most 3 (proof non-constructive)

15. (1+√3+ε)-approximation on k-median

16. k -median and UFL  f = cost of a facility  f #open facilities Given a black-box α-approximation A for UFL Naïve try : find an f such that A opens k facilities α-approxition for k-median? Proof : α ≈ 1.488 for UFL, α > 1.736 for k-median

17. k -median and UFL Naïve try : find an f such that A opens k facilities 2 issues with naïve try : 1. need LMP α-approximation for UFL α- approximation: LMP α-approximation LMP = Lagragean Multiplier Preserving

18. k -median and UFL S 1 : set of k 1 < k facilities S 2 : set of k 2 > k facilities bi-point solution Naïve try : find an f such that A opens k facilities 2 issues with naïve try : 1. need LMP α-approximation for UFL 2. can not find f s.t. A opens exactly k facilities

19. k -median and UFL 2 issues with naïve try : 1. need LMP α-approximation for UFL 2. can not find f s.t. A opens exactly k facilities LMP approx. factor bi-point  integral final ratio for k-median [JV] [JMS] 3 x 2 6 2 4 our result 2 do not know how to improve this factor of 2 is tight !!

20. bi-point solution k 1 = | S 1 | < k ≤ | S 2 | = k 2 a, b : ak 1 + bk 2 = k, a + b = 1 bi-point solution : a S 1 +b S 2 cost(a S 1 +b S 2 ) = a cost( S 1 ) + b cost( S 2 ) S1S1 S2S2

21. gap-2 instance 1 0 k + 1 cost of integral solution = 2 k 1 = 1, k 2 = k+1 cost ( S 1 ) = k+1, cost ( S 2 ) = 0 S1S1 S2S2

22. k -median and UFL Main Lemma 2 : bi-point solution of cost C  solution of cost with k+O(1/ε) facilities [JV][JMS]our result LMP approx. factor 322 bi-point  integral x 2 final ratio for k-median 64 this factor of 2 is tight !! bi-point  pseudo-integral Main Lemma 1 : suffice to give an α-approximate solution with k+O(1) facilities

23. Main Lemma 1 with k+1 open facilities, cost = 0 with k open facilities, cost huge A : black-box α-approximation with k+c open facilities A ' : (α+ε)-approximation with k open facilities A ' calls A n O(c/ε) times. bad instance:

24. Dense Facility B i : set of clients in a small ball around i i is A-dense, if connection cost of B i in OPT is ≥ A i BiBi this instance : i is A-dense for A ≈ opt

25. Dense Facility BiBi Reduction component works directly if there are no opt/t-dense facilities, t = O(c/ε) can reduce to such an instance in n O(t) time i

26. [Awasthi-Blum-Sheffet] : ε, δ >0 constants, OPT k-1 ≥ (1+δ)OPT k  can find (1+ε)-approximation Main Lemma 1 : suffice to give an α-approximate solution with k+O(1) facilities  k-median clustering is easy in practice  reason : there is a “meaningful” clustering Lemma 1 from [ABS]

27. Algorithm  Apply A to (k-c, F, C, d)  solution with k facilities of cost ≤ αOPT k-c  Apply [ABS] to each (k-i, F, C, d) for i = 0, 1, 2, …, c-1  Output the best of the c+1 solutions Proof  If OPT k-c ≤ (1+ε)OPT k, then done.  otherwise, consider the smallest i s.t. OPT k-i-1 ≥ (1+ε) 1/c OPT k-i  [ABS] on (k-i, F, C, d)  solution of cost (1+ε)OPT k-i ≤ (1+ε) 2 OPT k [ABS] OPT k-1 ≥ (1+δ)OPT k  (1+ε)-approximation A : α-approximation algorithm for k-median with k+c medians

28. Main Lemma 2 : bi-point solution of cost C  solution of cost with k+O(1/ε) facilities [JV] bi-point solution of cost C  solution of cost 2C  based on improving [JV] algorithm

29. S1S1 S2S2 given : bi-point solution a S 1 +b S 2 select S’ 2  S 2, | S’ 2 | = | S 1 | = k 1 with prob. a, open S 1 with prob. b, open S’ 2 randomly open k-k 1 facilities in S 2 \ S’ 2 i JV algorithm τ i = nearest facility of i guarantee : either i is open, or τ i is open

30. Analysis of JV algorithm i1i1 i2i2 i3i3 ≤ d 1 + d 2 If i 2 is open, connect j to i 2 Otherwise, if i 1 is open, connect j to i 1 Otherwise connect j to i 3 E[cost of j] ≤ × [cost of j in a S 1 +b S 2 ] d1d1 d2d2 j i 1  S 1, i 3  S’ 2 either i 1 or i 3 is open 2

31. Our Algorithm on average, d 1 >> d 2 d(j, i 3 ) ≤ i1i1 i2i2 i3i3 d1d1 d2d2 ≤ d 1 + d 2 j i3i3 If i 2 is open, connect j to i 2 Otherwise, if i 1 is open, connect j to i 1 Otherwise connect j to i 3 E[cost of j] ≤ × [cost of j in a S 1 +b S 2 ] 2 d 1 +2 d 2 2d1+d22d1+d2

32. Our Algorithm for a star, either the center is open, or all leaves are open idea : big stars: always open the center, open each leaf with prob. ≈b group small stars of the same size, dependent rounding for each group, open 3 more facilities than expected first try open each star independently? with prob. a, open the center, with prob. b, open the leaves problem : can not bound the number of open facilities need to guarantee : either i is open, or τ i is open i τiτi

33. small stars small star : star of size ≤ 2/(abε ) M h : set of stars of size h, m = |M h | Roughly, for am stars, open the center for bm stars, open the leaves More accurately, permute the stars and the facilities open top centers open bottom leaves

34. big stars size h > 2/(abε ) always open the center randomly open leaves ≈ bh for big star

35. Lemma : we open at most k + 6/(abε) facilities. for a big star of size h, FRAC : a+bh ALG : for a group of m small stars of size h FRAC : m(a+bh) ALG : there are at most 2/(abε) groups

36. Summary Main Lemma 2 : bi-point solution of cost C  solution of cost with k+O(1/ε) facilities [JV][JMS]our result LMP approx. factor 322 x 2 final ratio for k-median 64 bi-point  pseudo-integral Main Lemma 1 : suffice to give an α-approximate solution with k+O(1) facilities

37. Open Problems gap between integral solution with k+1 open facilities and LP value(with k open facilities)? tight analysis? algorithm works for k-means?