1. 1 Krakow, Jan. 9, 2008 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu

2. 2 Krakow, Jan. 9, 2008 Choose d 1 < d 2 < d 3 … (typically powers of 2) For j = 1, 2, 3, … Assume that the optimum is ≤ d j Using this bound attempt to construct a solution of cost ≤ C·d j Simple and effective Works for many offline and online problems Typically not best possible ratios Doubling method: (for a minimization problem)

3. 3 Krakow, Jan. 9, 2008 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 Krakow, Jan. 9, 2008 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 Krakow, Jan. 9, 2008 Online Bidding 1 2 5 12

6. 6 Krakow, Jan. 9, 2008 Online Bidding 1 2 5 12 20 bags of gunpowder but… 6 bags could have been enough so ratio = 20/6

7. 7 Krakow, Jan. 9, 2008 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 Krakow, Jan. 9, 2008 Deterministic Bidding - Upper Bound If 2 j-1 < u ≤ 2 j, the ratio is Doubling strategy: bid 1, 2, 4, …, 2 i, …

9. 9 Krakow, Jan. 9, 2008 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]

10. 10 Krakow, Jan. 9, 2008 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

11. 11 Krakow, Jan. 9, 2008 d1d1 d2d2 d3d3 d4d4 0 u Cow-Path

12. 12 Krakow, Jan. 9, 2008 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

13. 13 Krakow, Jan. 9, 2008 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] …

14. 14 Krakow, Jan. 9, 2008 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

15. 15 Krakow, Jan. 9, 2008 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)

16. 16 Krakow, Jan. 9, 2008 customer facility (potential)

17. 17 Krakow, Jan. 9, 2008 k = 2 facilities 3 1 22 3 1 4 1 cost = 17

18. 18 Krakow, Jan. 9, 2008 k = 4 facilities 1 2 2 1 1 1 1 3 cost = 12

19. 19 Krakow, Jan. 9, 2008 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)

20. 20 Krakow, Jan. 9, 2008 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

21. 21 Krakow, Jan. 9, 2008 k = 1 2 4 5 3 2 2 5 3 cost = 26 opt = 26

22. 22 Krakow, Jan. 9, 2008 k = 2 2 1 2 2 2 2 4 3 cost = 18 !!! opt = 17

23. 23 Krakow, Jan. 9, 2008 k = 2 2 1 2 2 2 1 4 1 cost = 15 opt = 17

24. 24 Krakow, Jan. 9, 2008 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 …

25. 25 Krakow, Jan. 9, 2008 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

26. 26 Krakow, Jan. 9, 2008 Theorem: The optimal size-competitive ratio for incremental medians is: 4 in the deterministic case e ≈ 2.72 in the randomized case [Chrobak, Kenyon, Noga, Young, ‘06]

27. 27 Krakow, Jan. 9, 2008 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

28. 28 Krakow, Jan. 9, 2008 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 …)

29. 29 Krakow, Jan. 9, 2008 Example: Star with m arms, w farmers per cluster 1 0 1 1

30. 30 Krakow, Jan. 9, 2008 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

31. 31 Krakow, Jan. 9, 2008 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  ∞

32. 32 Krakow, Jan. 9, 2008 Cost-Competitive Incremental Medians [Mettu, Plaxton ‘00]: Lower bound of 2 Upper bound C ≈ 30 (in polynomial time) use doubling to improve to 8

33. 33 Krakow, Jan. 9, 2008 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”

34. 34 Krakow, Jan. 9, 2008 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)

35. 35 Krakow, Jan. 9, 2008 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)

36. 36 Krakow, Jan. 9, 2008 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(0), 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

37. 37 Krakow, Jan. 9, 2008 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

38. 38 Krakow, Jan. 9, 2008 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)

39. 39 Krakow, Jan. 9, 2008 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

40. 40 Krakow, Jan. 9, 2008 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

41. 41 Krakow, Jan. 9, 2008 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]