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A Primal-Dual Approach to Online Optimization Problems

A Primal-Dual Approach to Online Optimization Problems

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A Primal-Dual Approach to Online Optimization Problems

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  1. A Primal-Dual Approach to Online Optimization Problems

  2. Online Optimization Problems • input arrives “piece by piece” (“piece” is called request) • upon arrival of a request - has to be served immediately • past decisions cannot be revoked how to evaluate the performance of an online algorithm? • if, for each request sequence, cost(online) ≤ r x cost(optimal offline) • then online algorithm is r-competitive

  3. Road Map Introducing the framework: • Ski rental • Online set cover • Virtual circuit routing The general framework: • {0,1} covering/packing linear programs • General covering/packing linear programs Recent results: • The ad-auctions problem • Weighted caching

  4. The Ski Rental Problem • Buying costs $B. • Renting costs $1 per day. Problem: • Number of ski days is not known in advance. Goal: Minimize the total cost.

  5. Ski Rental – Integer Program Subject to: For each day i:

  6. Ski Rental – Relaxation D: Dual Packing For each day i: P: Primal Covering For each day i: • Online setting: • Primal: New constraints arrive one by one. • Requirement: Upon arrival, constraints should be satisfied. • Monotonicity: Variables can only be increased.

  7. Ski Rental – Algorithm D: Dual Packing For each day i: P: Primal Covering For each day i: • Initially x 0 • Each new day (new constraint): • if x<1: • zi 1-x • x  x(1+ 1/B) + 1/(c*B) - ‘c’ later. • yi  1

  8. Analysis of Online Algorithm Proof of competitive factor: • Primal solution is feasible. • In each iteration, ΔP ≤ (1+ 1/c)ΔD. • Dual is feasible. Conclusion: Algorithm is (1+ 1/c)-competitive 1-2-3 • Initially x 0 • Each new day (new constraint): • if x<1: • zi 1-x • x  x(1+ 1/B) + 1/(c*B) - ‘c’ later. • yi  1

  9. Analysis of Online Algorithm • Primal solution is feasible. If x ≥1 the solution is feasible. Otherwise set: zi 1-x. • In each iteration, ΔP ≤ (1+ 1/c)ΔD: If x≥1, ΔP =ΔD=0 Otherwise: • Change in dual: 1 • Change in primal: BΔx + zi = x+ 1/c+ 1-x = 1+1/c Algorithm: When new constraint arrives, if x<1: zi1-x x x(1+ 1/B) + 1/c*B yi1

  10. Analysis of Online Algorithm • Dual is feasible: Need to prove: We prove that after B days x≥1 x is a sum of geometric sequence a1 = 1/(cB), q = 1+1/B Algorithm: When new constraint arrives, if x<1: zi1-x x x(1+ 1/B) + 1/c*B yi1

  11. Randomized Algorithm 0 1 • Choose d uniformly in [0,1] • Buy on the day corresponding to the “bin” d falls in • Rent up to that day Analysis: • Probability of buying on the i-th day is xi • Probability of renting on the i-th day is at most zi X: X1 X2 X3 X4

  12. Key Idea for Primal-Dual Primal: Min i ci xi Dual: Max t bt yt Step t, newconstraint:New variable yt a1x1 + a2x2 + … + ajxj≥ bt + bt yt in dual objective xi (1+ ai/ci) xi(mult. update)yt  yt + 1 (additive update) primal cost = = Dual Cost

  13. The Online Set-Cover Problem • Elements: e1, e2, …, en • Set system: s1, s2, … sm • Costs: c(s1), c(s2), … c(sm) Online Setting: • Elements arrive one by one. • Upon arrival elements need to be covered. • Sets that are chosen cannot be “unchosen”. Goal: Minimize the cost of the chosen sets.

  14. Set Cover – Linear Program D: Dual Packing P: Primal Covering • Online setting: • Primal: constraints arrive one by one. • Requirement: each constraint is satisfied. • Monotonicity: variables can only be increased.

  15. Set Cover – Algorithm D: Dual Packing P: Primal Covering • Initially x(s) 0 • When new element arrives, while • y(e)  y(e)+1 • .

  16. Analysis of Online Algorithm Proof of competitive factor: • Primal solution is feasible. • In each iteration, ΔP ≤ 2ΔD. • Dual is (almost) feasible. Conclusion: We will see later. 1-2-3 • Initially x(S) 0 • When new element e arrives, while • y(e)  y(e)+1 • .

  17. Analysis of Online Algorithm • Primal solution is feasible. We increase the primal variables until the constraint is feasible. • Initially x(S) 0 • When new element e arrives, while • y(e)  y(e)+1 • .

  18. Analysis of Online Algorithm • In each iteration, ΔP ≤ 2ΔD. In each iteration: • ΔD = 1 • Initially x(S) 0 • When new element e arrives, while • y(e)  y(e)+1 • .

  19. Analysis of Online Algorithm • Dual is (almost) feasible: • We prove that: • If y(e) increases, then x(s) increases (for e in S). • x(s) is a sum of a geometric series: a1 = 1/[mc(s)], q = (1+ 1/c(s)) • Initially x(S) 0 • When new element e arrives, while • y(e)  y(e)+1 • .

  20. Analysis of Online Algorithm • After c(s)O(log m) rounds: We never increase a variable x(s)>1! • Initially x(S) 0 • When new element e arrives, while • y(e)  y(e)+1 • .

  21. Conclusion • The dual is feasible with cost 1/O(log m) of the primal. • The algorithm produces a fractional set cover that is O(log m)-competitive. • Remark: No online algorithm can perform better in general. What about an integral solution? • Round fractional solution. (With O(log n) amplification.) • Can be done deterministically online [AAABN03]. • Competitive ratio is O(log m log n).

  22. Online Virtual Circuit Routing Network graph G=(V, E) capacity function u: E Z+ Requests: ri = (si, ti) • Problem: Connect si to ti by a path, or reject the request. • Reserve one unit of bandwidth along the path. • No re-routing is allowed. • Load: ratio between reserved edge bandwidth and edge capacity. • Goal: Maximize the total throughput.

  23. Routing – Linear Program = Amount of bandwidth allocated for ri on path p s.t: For each ri: For each edge e: - Available paths to serve request ri

  24. Routing – Linear Program D: Dual Packing P: Primal Covering • Online setting: • Dual: new columns arrive one by one. • Requirement: each dual constraint is satisfied. • Monotonicity: variables can only be increased.

  25. Routing – Algorithm D: Dual Packing P: Primal Covering • Initially x(e) 0 • When new request arrives, if • z(ri)  1 • . • y(ri,p)  1

  26. Analysis of Online Algorithm Proof of competitive factor: • Primal solution is feasible. • In each iteration, ΔP ≤ 3ΔD. • Dual is (almost) feasible. Conclusion: We will see later. 1-2-3 • Initially x(e) 0 • When new request arrives, if • z(ri)  1 • . • y(ri,p)  1

  27. Analysis of Online Algorithm • Primal solution is feasible. If the solution is feasible. Otherwise: we update z(ri)  1 • Initially x(e) 0 • When new request arrives, if • z(ri)  1 • . • y(ri,p)  1

  28. Analysis of Online Algorithm • In each iteration: ΔP ≤ 3ΔD. If ΔP = ΔD=0 Otherwise: ΔD=1 • Initially x(e) 0 • When new request arrives, if • z(ri)  1 • . • y(ri,p)  1

  29. Analysis of Online Algorithm • Dual is (almost) feasible. We prove: • For each e, after routing u(e)O(log n) on e, x(e)≥1 x(e) is a sum of a geometric sequence x(e)1 = 1/(nu(e)), q = 1+1/u(e) • After u(e)O(log n) requests:

  30. New Results via P-D Approach: Routing Previous results (routing/packing): • [AAP93] – Route O(log n) fraction of the optimal without violating capacity constraints. Capacities must be at least logarithmic. • [AAFPW94] – Route all the requests with load of at most O(log n) times the optimal load. Observation [BN06] – Both results can be described within the primal-dual approach.

  31. New Results via P-D Approach: Routing We saw a simple algorithm which is: • 3-competitive and violates capacities by O(log n) factor. Can be improved [Buchbinder, Naor, FOCS06] to: • 1-competitive and violates capacities by O(log n) factor. Non Trivial. Main ideas: • Combination of ideas drawn from casting of previous routing algorithms within the primal-dual approach. • Decomposition of the graph. • Maintaining several primal solutions which are used to bound the dual solution, and for the routing decisions.

  32. New Results via P-D Approach: Routing Applications [Buchbinder, N, FOCS 06]: • Can be used as “black box” for many objective functions and in many routing models: • Previous Settings [AAP93,APPFW94]. • Maximizing throughput. • Minimizing load. • Achieving better global fairness results (Coordinate competitiveness).

  33. Road Map Introducing the framework: • Ski rental • Online set cover • Virtual circuit routing The general framework: • {0,1} covering/packing linear programs • General covering/packing linear programs Recent results: • The ad-auctions problem • Weighted caching

  34. Online Primal-Dual Approach • Can the offline problem be cast as a linear covering/packing program? • Can the online process be described as: • New rows appearing in a covering LP? • New columns appearing in a packing LP? Yes ?? • Upon arrival of a new request: • Update primal variables in a multiplicative way. • Update dual variables in an additive way.

  35. Online Primal Dual Approach Next Prove: • Primal solution is feasible (or nearly feasible). • In each round, ΔP ≤ c ΔD. • Dual is feasible (or nearly feasible). Got a fractional solution, but need an integral solution ?? • Randomized rounding techniques might work. • Sometimes, even derandomization (e.g., method of conditional probabilities) can be applied online! 1-2-3

  36. Online Primal-Dual Approach Advantages: • Generic ideas and algorithms applicable to many online problems. • Linear Program helps detecting the difficulties of the online problem. • General recipe for the design and analysis of online algorithms. • No potential function appearing “out of nowhere”. • Competitiveness with respect to a fractional optimal solution.

  37. General Covering/Packing Results What can you expect to get? • For a {0,1} covering/packing matrix: • Competitive ratio O(log D) [BN05] (D – max number of non-zero entries in a constraint). Remarks: • Fractional solutions. • Number of constraints/variables can be exponential. • There can be a tradeoff between the competitive ratio and the factor by which constraints are violated.

  38. General Covering/Packing Results • For a general covering/packing matrix [BN05] : Covering: • Competitive ratio O(log n) (n – number of variables). Packing: • Competitive ratio O(log n + log [a(max)/a(min)]) a(max), a(min) – maximum/minimum non-zero entry Remarks: • Results are tight.

  39. Special Cases The max number of non-zero entries in a constraint is a constant? • You can get a constant ratio. The max number of non-zero entries in a constraint is 2? • Calls for an e/(e-1)-ratio. Examples: • Ski rental, Online matching, Ad-Auctions.

  40. Known Results via P-D Approach Covering Online Problems (Minimization): • O(log k)-algorithm for weighted caching [BBN07] • Ski rental, Dynamic TCP Acknowledgement • Parking Permit Problem [Meyerson 05] • Online Set Cover [AAABN03] • Online Graph Covering Problems [AAABN04]: • Non-metric facility location • Generalized connectivity: pairs arrive online • Group Steiner: groups arrive online • Online multi-cut: (s,t)--pairs arrive online

  41. Known Results via P-D Approach Packing Online Problems (maximization): • Online Routing/Load Balancing Problems [AAP93, AAPFW93, BN06]. • General Packing/routing e.g. Multicast trees. • Online Matching [KVV91]–Nodes arrive one-by-one. • Ad-Auctions Problem [MSVV05]– In a bit …

  42. Road Map Introducing the framework: • Ski rental • Online set cover • Virtual circuit routing The general framework: • {0,1} covering/packing linear programs • General covering/packing linear programs Recent results: • The ad-auctions problem • Weighted caching

  43. What are Ad-Auctions? Vacation Eilat You type in a search engine: You get: And … Advertisements Algorithmic Search results

  44. How do search engines sell ads? • Each advertiser: • Sets a daily budget • Provides bids on interesting keywords • Search Engine (on each keyword): • Selects ads • Advertiser pays bid if user clicks on ad. Goal (of Search engine): Maximize revenue

  45. Mathematical Model • Buyer i: • Has a daily budget B(i) • Online Setting: • Items (keywords) arrive one-by-one. • Each buyer gives a bid on each of the items (can be zero) • Algorithm: • Assigns each item to some interested buyer. Assumption: Bids are small compared to the daily budget.

  46. Ad-Auctions – Linear Program I - Set of buyers. J - Set of items. B(i) – Budget of buyer i b(i,j) – bid of buyer i on item j s.t: For each item j: For each buyer i: = 1 j-th ad-auction is sold to buyer i. Each item is sold once. Buyers do not exceed their budget

  47. Results [MSVV FOCS 05]: • (1-1/e)-competitive online algorithm. • Bound is tight. • Analysis uses tradeoff revealing family of LP’s -not very intuitive. Our Results [Buchbinder, Jain, N, 2007]: • A different approach based on the primal-dual method: very simple and intuitive … and extensions. • Techniques are applicable to many other problems.

  48. The Paging/Caching Problem Set of n pages, cache of size k<n. Request sequence of pages1, 6, 4, 1, 4, 7, 6, 1, 3, … If requested page is in cache, nopenalty. Else, cache miss! And load page into cache, (possibly) evicting some page. Goal: Minimize the number of cache misses. Main Question: Which page to evacuate?

  49. Previous Results: Paging Paging (Deterministic) [Sleator Tarjan 85]: • Any det. algorithm >= k-competitive. • LRU is k-competitive (also other algorithms) • LRU is k/(k-h+1)-competitive if optimal has cache of size h<k Paging (Randomized): • Rand. Marking O(log k) [Fiat, Karp, Luby, McGeoch, Sleator, Young 91]. • Lower bound Hk [Fiat et al. 91], tight results known. • O(log(k/k-h+1))-competitive algorithm if optimal hascache of size h<k[Young 91]

  50. The Weighted Paging Problem One small change: • Each page i has a different load cost w(i). • Models scenarios in which the cost of bringing pages is not uniform: Main memory, disk, internet … Goal • Minimize the total cost of cache misses. web