1. 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu

2. 2 Wroclaw University, Sept 18, 2007 Doubling method: (for a minimization problem) Choose d 1 < d 2 < d 3 … (typically powers of 2) For j = 1, 2, 3, … Assume that the optimum is ≤ d j Use this bound to construct a solution of cost ≤ C·d j Simple and effective (works for many problems, offline and online) Typically not best possible ratios

3. 3 Wroclaw University, Sept 18, 2007 Outline: 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering

4. 4 Wroclaw University, Sept 18, 2007 Outline: 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering

5. 5 Wroclaw University, Sept 18, 2007 Online Bidding 1 2 5 12

6. 6 Wroclaw University, Sept 18, 2007 Online Bidding 1 2 5 12 20 bags of gunpowder but… 6 bags could have been enough so ratio = 20/6

7. 7 Wroclaw University, Sept 18, 2007 Online Bidding Item for sale of value u (unknown to bidder) Buyer bids d 1,d 2,d 3, … until some d j ≥ u Cost: d 1 + d 2 + … + d j Optimum = u Competitive ratio

8. 8 Wroclaw University, Sept 18, 2007 Deterministic Bidding - Upper Bound If 2 j-1 < u ≤ 2 j, the ratio is Doubling strategy: bid 1, 2, 4, …, 2 i, …

9. 9 Wroclaw University, Sept 18, 2007 Deterministic Bidding - Lower Bound Proof idea: Suppose R < 4. For u = d j-1 +ε d 1 + d 2 + … + d j ≤ R·u  R·d j-1 Taking s i = d 1 + … + d i, this is s j ≤ R (s j-1 – s j-2 ) In “worst case”, we get s j - R s j-1 + R s j-2 = 0 R < 4  characteristic equation has no real roots  {s i } is not monotonely increasing

10. 10 Wroclaw University, Sept 18, 2007 Intuition 1 -- 4 = 2 · 2 (two factors of 2): first 2: sum of the geometric sequence second 2: overestimating adversary cost Intuition 2 -- after a bid d j, either Online Bidding - Intuitions we lose but …. we learn the optimum is large, that is u > d j we win

11. 11 Wroclaw University, Sept 18, 2007 Online Bidding Theorem: The optimal competitive ratio for online bidding is: 4 in the deterministic case e  2.72 in the randomized case Randomized e-ing strategy: choose uniformly random x  [0,1), and bid e x, e x+1, e x+2, e x+3, … [folklore] [Chrobak, Kenyon, Noga, Young, ‘06]

12. 12 Wroclaw University, Sept 18, 2007 Outline: 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering

13. 13 Wroclaw University, Sept 18, 2007 d1d1 d2d2 d3d3 d4d4 0 u Cow-Path

14. 14 Wroclaw University, Sept 18, 2007 Analysis: d1d1 d2d2 d3d3 d j+1 0 u d j-1 djdj For d j-1 < u ≤ d j+1 (j odd) 2  bidding ratio extra ratio 1 So the ratio = 2  bidding ratio + 1 = 9 for d j = 2 j

15. 15 Wroclaw University, Sept 18, 2007 For d j-1 < u ≤ d j+1 online bidding ratio

16. 16 Wroclaw University, Sept 18, 2007 Theorem: The optimal competitive ratio for the cow- path problem is 9 in the deterministic case  4.59 in the randomized case Solution of (r-1)ln(r-1) = r  2e+1 Connection to online bidding does not work in randomized case -- why? [Gal ‘80] [Baeza-Yates, Culberson, Rawlins ‘93] [Papadimitriou, Yannakakis ‘91] [Kao, Reif, Tate ‘94] …

17. 17 Wroclaw University, Sept 18, 2007 Outline: 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering

18. 18 Wroclaw University, Sept 18, 2007 The k-Median Problem X = set of facilities Y = set of customers X  Y : metric space with distance function d xy For F  X let cost(F) =  y  Y d yF where d yF = min f  F d yf The k-Median Problem: Find a facility set F of size k for which cost(F) is minimized. optimal F = Q k (the k-median)

19. 19 Wroclaw University, Sept 18, 2007 customer facility (potential)

20. 20 Wroclaw University, Sept 18, 2007 k = 2 facilities 3 1 22 3 1 4 1 cost = 17

21. 21 Wroclaw University, Sept 18, 2007 k = 4 facilities 1 2 2 1 1 1 1 3 cost = 12

22. 22 Wroclaw University, Sept 18, 2007 Offline Case k-Median is NP-hard Offline approximations: given k, find F such that |F | ≤ k and cost(F) ≤ C·opt k C-cost-approximation Upper bound C = 3+  [Arya, Garg, Khandekar, Munagala, Pandit ‘01] C ≥ 1+2/e for polynomial algorithms (unless P = NP) [Jain, Mahdian, Saberi ‘02] cost(F) ≤ opt k and |F| ≤ S·k S-size-approximation S = Ω(logn) for polynomial algorithms (unless P = NP)

23. 23 Wroclaw University, Sept 18, 2007 Size-Competitive Incremental Medians k not known, authorizations for additional facilities arrive over time Algorithm produces a sequence of facility sets: F 1  F 2  …  F n An algorithm is S-size-competitive if |F k | ≤ S·k and cost(F k ) ≤ opt k for all k. Goal: small competitive ratio

24. 24 Wroclaw University, Sept 18, 2007 k = 1 2 4 5 3 2 2 5 3 cost = 26 opt = 26

25. 25 Wroclaw University, Sept 18, 2007 k = 2 2 1 2 2 2 2 4 3 cost = 18 !!! opt = 17

26. 26 Wroclaw University, Sept 18, 2007 k = 2 2 1 2 2 2 1 4 1 cost = 15 opt = 17

27. 27 Wroclaw University, Sept 18, 2007 Size-Competitive Incremental Medians Algorithm: 1. choose d 1 < d 2 < d 3 … 2. Compute Q 1, Q 2, … (optimal medians) 3. F 1 = Q d(1) // d(j) = d j for k = 2, 3, … if k = d i +1 F k = F k-1  Q d(i+1) … not a polynomial time algorithm …

28. 28 Wroclaw University, Sept 18, 2007 Q d(1) Q k = optimal k-median Q d(4) 1 2 3 4 5 … 11 12 13 … 19 20 21 … 31 32 … d 1 d 2 d 3 d 4 k k kk Q d(2) Q d(3)

29. 29 Wroclaw University, Sept 18, 2007 Q d(2) 1 2 3 4 5 … 11 12 13 … 19 20 21 … 31 32 … d 1 d 2 d 3 d 4 kk k Q d(1) Q d(3) Q d(4) Q k = optimal k-median

30. 30 Wroclaw University, Sept 18, 2007 Q d(3) Q d(2) 1 2 3 4 5 … 11 12 13 … 19 20 21 … 31 32 … d 1 d 2 d 3 d 4 k Q d(1) k k Q d(4) Q k = optimal k-median

31. 31 Wroclaw University, Sept 18, 2007 Q d(4) 1 2 3 4 5 … 11 12 13 … 19 20 21 … 31 32 … d 1 d 2 d 3 d 4 k Q d(3) Q d(2) Q d(1) Q k = optimal k-median

32. 32 Wroclaw University, Sept 18, 2007 Analysis: At step k, for d j-1 < k ≤ d j cost(F k ) ≤ cost(Q d(j) ) = opt( d j ) ≤ opt k |F k | ≤ d 1 +d 2 + … + d j So the ratio is Same as online bidding So we get ratio = 4 for d j = 2 j

33. 33 Wroclaw University, Sept 18, 2007 Theorem: The optimal size-competitive ratio for incremental medians is: 4 in the deterministic case e ≈ 2.72 in the randomized case (Lower bound: prove that online bidding reduces to incremental medians) [Chrobak, Kenyon, Noga, Young, ‘06]

34. 34 Wroclaw University, Sept 18, 2007 Outline: 1. Online bidding 2. Cow-path 3. Incremental medians (size approximation) 4. Incremental medians (cost approximation) 5. List scheduling on related machines 6. Minimum latency tours 7. Incremental clustering

35. 35 Wroclaw University, Sept 18, 2007 Cost-Competitive Incremental Medians k not known, authorizations for additional facilities arrive over time Algorithm produces a sequence of facility sets: F 1  F 2  …  F n An algorithm is C-cost-competitive if |F k |≤ k and cost(F k ) ≤ C·opt k for all k. Goal: small competitive ratio (in polynomial time, if possible …)

36. 36 Wroclaw University, Sept 18, 2007 Example: Star with m arms, w farmers per cluster 1 0 1 1

37. 37 Wroclaw University, Sept 18, 2007 Example: Star with m arms, w farmers per cluster 1 0 1 1 k = 1 cost = 2(m-1)w ≈ 2  opt cost So C  2

38. 38 Wroclaw University, Sept 18, 2007 Example: Star with m arms, w farmers per cluster 1 0 1 1 k = 1 cost = w opt cost = 0 2 3 4 … m So C  ∞

39. 39 Wroclaw University, Sept 18, 2007 Cost-Competitive Incremental Medians [Mettu, Plaxton ‘00]: Lower bound of 2 Upper bound C ≈ 30 (in polynomial time) use doubling to improve to 8

40. 40 Wroclaw University, Sept 18, 2007 Idea: construct sequence backwards, at each step extracting next set from previous one for k’ < k we want to show that F k contains a cheap subset F k’ customers facilities FkFk F k’ F k”

41. 41 Wroclaw University, Sept 18, 2007 Lemma: F, Q facility sets. |F| = k H |Q| = k’ < k H = H(Q,F) = k’ facilities in F closest to the points in Q Then cost(H)  cost(F) + 2·cost(Q)

42. 42 Wroclaw University, Sept 18, 2007 Proof: Choose H Q F customer x f h q f  F : closest to x q  Q : closest to x h  H : closest to q (in F) d xH ≤ d xh ≤ d xq + d qh ≤ d xq + d qf ≤ d xq + (d xf + d xq ) = 2d xq + d xf = 2d xQ + d xF So cost(H) ≤ 2·cost(Q) + cost(F)

43. 43 Wroclaw University, Sept 18, 2007 Algorithm: 1. Choose d 1 < d 2 < d 3 < … Wlog. opt n = cost(X) = 1 2. Choose p(1) > … > p(m) = 1 s.t. cost(Q p(i) ) = opt p(i) = d i (For simplicity assume they exist) 3. Construct sets F k for k = n, p(1), p(2),… F n  X (all facilities) F p(i+1)  H (F p(i), Q p(i+1) ) for i= 2,…,m 4. For p(i+1) < k < p(i) set F k  F p(i+1) (So for these k we have |F k | ≤ k) 5. Output F 1, F 2,…, F n

44. 44 Wroclaw University, Sept 18, 2007 Constructing F p(i) cost(F p(i) ) ≤ cost(F p(i-1) ) +2·cost(Q p(i) ) ≤ cost(F p(i-1) ) +2·d i H Qp(i)Qp(i) Fp(i) Fp(i) F p(i-1) Q k = optimal k-median

45. 45 Wroclaw University, Sept 18, 2007 F p(i-2) cost(F p(i) ) ≤ cost(F p(i-1) ) + 2·d i ≤ cost(F p(i-2) ) + 2·d i-1 + 2·d i ≤ … ≤ 2 · (d 1 + d 2 + …. + d i ) Analysis: F p(i-1) Fp(i) Fp(i) Q p(i-1) optimal Q p(i)

46. 46 Wroclaw University, Sept 18, 2007 Suppose p(j) < k ≤ p(j-1) Then opt k ≥ opt p(j-1) = d j-1 cost(F k ) = cost(F p(j) ) ≤ 2 · (d 1 + d 2 + …. + d j ) This is 2  (bidding ratio) So we get ratio = 8 for d j = 2 j

47. 47 Wroclaw University, Sept 18, 2007 Theorem: Upper bounds for cost-competitive incremental medians: Deterministic 8 24+  in polynomial time Randomized 2e 6e +  ≈ 16.31 +  in polynomial time [Lin, Nagarajan, Rajamaran, Williamson ‘06] [Chrobak, Kenyon, Noga, Young ‘06] Use (3+  )-approximate medians instead of optimal ones

48. 48 Wroclaw University, Sept 18, 2007 Current world records: 16+ , deterministic polynomial time 4e + , randomized polynomial time [Lin, Nagarajan, Rajamaran, Williamson ‘06] Deterministic (not polynomial-time) Lower bound of 2.0013 Upper bound of 7.65 [Chrobak, Hurand ‘07]