Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems

Transcription

1 Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems Pierre Fraigniaud Magnús M. Halldórsson Boaz Patt-Shamir CNRS, U. Paris Diderot Dror Rawitz Tel Aviv U. Reykjavik U. Adi Rosén CNRS, U. Paris Diderot Tel Aviv U. Narrated by: Moti Medina (Tel Aviv U.) Shrinking Maxima, Decreasing Costs APPROX / 15

2 Motivation: Video Transmission Over Networks Large Data Items Sequence consisting of large frames (I-frames: SD hundreds of Kb; HD several Mb) Small Transfer Units IP packet size 64Kb Practically 1.5Kb (Ethernet) Shrinking Maxima, Decreasing Costs APPROX / 15

3 Motivation: Video Transmission Over Networks Large Data Items Sequence consisting of large frames (I-frames: SD hundreds of Kb; HD several Mb) Small Transfer Units IP packet size 64Kb Practically 1.5Kb (Ethernet) Link Ingresses Problem Packets arrive in bursts at an outgoing link of a router In case of an overflow: which packet to serve? to drop? If packets are dropped, the whole frame may be lost! Shrinking Maxima, Decreasing Costs APPROX / 15

4 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Shrinking Maxima, Decreasing Costs APPROX / 15

5 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 1 Shrinking Maxima, Decreasing Costs APPROX / 15

6 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: Dropped: F 1 Shrinking Maxima, Decreasing Costs APPROX / 15

7 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 5 F 2 Dropped: F 1 Shrinking Maxima, Decreasing Costs APPROX / 15

8 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 Dropped: F 1 F 5 Shrinking Maxima, Decreasing Costs APPROX / 15

9 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 6 F 2 F 1 Dropped: F 1 F 5 Shrinking Maxima, Decreasing Costs APPROX / 15

10 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 Dropped: F 1 F 5 F 1 Shrinking Maxima, Decreasing Costs APPROX / 15

11 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 5 F 2 F 6 F 2 Dropped: F 1 F 5 F 1 Shrinking Maxima, Decreasing Costs APPROX / 15

12 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 Dropped: F 1 F 5 F 1 F 2 F 5 Shrinking Maxima, Decreasing Costs APPROX / 15

13 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 6 F 4 F 2 F 6 Dropped: F 1 F 5 F 1 F 2 F 5 Shrinking Maxima, Decreasing Costs APPROX / 15

14 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 Dropped: F 1 F 5 F 1 F 2 F 4 F 5 F 6 Shrinking Maxima, Decreasing Costs APPROX / 15

15 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 F 1 Dropped: F 1 F 5 F 1 F 2 F 4 F 5 F 6 Shrinking Maxima, Decreasing Costs APPROX / 15

16 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 4 F 2 F 6 F 1 F 2 Dropped: F 1 F 5 F 1 F 2 F 4 F 5 F 6 Shrinking Maxima, Decreasing Costs APPROX / 15

17 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 F 1 F 2 Dropped: F 1 F 5 F 1 F 2 F 4 F 4 F 5 F 6 Shrinking Maxima, Decreasing Costs APPROX / 15

18 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 F 1 F 2 F 5 Dropped: F 1 F 5 F 1 F 2 F 4 F 4 F 5 F 6 Shrinking Maxima, Decreasing Costs APPROX / 15

19 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 F 1 F 2 F 5 alg = { } Dropped: F 1 F 5 F 1 F 2 F 4 F 4 F 5 F 6 Shrinking Maxima, Decreasing Costs APPROX / 15

20 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 F 1 F 2 alg = { } F 5 opt = {F 1, F 4, F 5 } Dropped: F 1 F 5 F 1 F 2 F 4 F 4 F 5 F 6 Shrinking Maxima, Decreasing Costs APPROX / 15

21 Multi-Part Tasks Online Scenario: Bursts arrive one by one Each packet belongs to some data item Upon arrival a burst, we transmit one packet, and the rest are dropped Number of bursts and number of data items are unknown Example: F 2 F 6 F 1 F 2 alg = { } F 5 opt = {F 1, F 4, F 5 } Dropped: F 1 F 5 F 1 F 2 F 4 F 4 F 5 F 6 x 2 + x 5 1 x 3 + x 4 + x 6 1 Shrinking Maxima, Decreasing Costs APPROX / 15

22 Online Packing Integer Programs (OPIP) Packing Integer Program: a ij ith requirement of item j b j benefit for item j c i capacity of ith constraint p j upper bound on # copies of j max s.t. n j=1 b jx j n j=1 a ijx j c i x j p j x j N i j j Shrinking Maxima, Decreasing Costs APPROX / 15

23 Online Packing Integer Programs (OPIP) Packing Integer Program: a ij ith requirement of item j b j benefit for item j c i capacity of ith constraint p j upper bound on # copies of j max s.t. n j=1 b jx j n j=1 a ijx j c i x j p j x j N i j j Online PIP: Init: x j = p j, for all i Knapsack constraints (rows) arrive one by one Variables may only decrease Shrinking Maxima, Decreasing Costs APPROX / 15

24 Online Packing Integer Programs (OPIP) Packing Integer Program: a ij ith requirement of item j b j benefit for item j c i capacity of ith constraint p j upper bound on # copies of j max s.t. n j=1 b jx j n j=1 a ijx j c i x j p j x j N i j j Online PIP: Init: x j = p j, for all i Knapsack constraints (rows) arrive one by one Variables may only decrease Standard Online Model: [Awerbuch Azar Plotkin 93], [Buchbinber Naor 09] Columns arrive one by one Upon arrival of a column, algorithm may accept or reject Variables may increase Shrinking Maxima, Decreasing Costs APPROX / 15

25 OPIP Related Work Special Cases: Independent Set Set Packing k-dimensional Matching Not approximable within O( k log k ) [Hazan et al. 03] Shrinking Maxima, Decreasing Costs APPROX / 15

26 OPIP Related Work Special Cases: Independent Set Set Packing k-dimensional Matching Not approximable within O( k log k ) [Hazan et al. 03] Competitive ratios depend on Max column: C = max j Pi a ij Max row: R = max i Pj a ij Max overload ratio: ρ = max i 1 c i Pj a ij Shrinking Maxima, Decreasing Costs APPROX / 15

27 OPIP Related Work Special Cases: Independent Set Set Packing k-dimensional Matching Not approximable within O( k log k ) [Hazan et al. 03] Competitive ratios depend on Max column: C = max j Pi a ij Max row: R = max i Pj a ij Max overload ratio: ρ = max i 1 c i Pj a ij Scheduling with Interval Conflicts: [Halldórsson Patt-Shamir Rawitz 11] Consecutive ones constraints Ω(log R) (centralized) and O(log R) (distributed) Shrinking Maxima, Decreasing Costs APPROX / 15

28 OPIP Previous and Our Results Online Set Packing [Emek Halldórsson Mansour Patt-Shamir Radhakrishnan Rawitz 10] A = {0, 1} m n, c = 1 m, p = 1 n Deterministic Lower Bound: R C 1 Randomized Lower Bound: Ω Centralized C R in construction Randomized Upper Bound: C R Distributed Extends to capacitated OSP: 2C ρ ( C R ( log log C log C ) 2 ) Shrinking Maxima, Decreasing Costs APPROX / 15

29 OPIP Previous and Our Results Online Set Packing [Emek Halldórsson Mansour Patt-Shamir Radhakrishnan Rawitz 10] A = {0, 1} m n, c = 1 m, p = 1 n Deterministic Lower Bound: R C 1 Randomized Lower Bound: Ω Centralized C R in construction Randomized Upper Bound: C R Distributed Extends to capacitated OSP: 2C ρ ( C R ( log log C log C ) 2 ) Our Result #1: Extend OSP upper bound to OPIP p = 1 n : 2C ρ-competitive algorithm Similar, but not the same algorithm p 1 n : Additional max j p j factor Shrinking Maxima, Decreasing Costs APPROX / 15

30 Randomized Algorithm for OPIP Special case: b = 1 n unweighted p = 1 n single copy per item c = u n uniform capacities Shrinking Maxima, Decreasing Costs APPROX / 15

31 Randomized Algorithm for OPIP Special case: b = 1 n unweighted p = 1 n single copy per item c = u n uniform capacities Algorithm: For each item j pick a random priority r(j) U[0, 1] Upon arrival of a constraint: Construct u subsets S i1,...,s iu Choose a ij subsets at random for item j For each subset S il reject all items but the one with highest priority Shrinking Maxima, Decreasing Costs APPROX / 15

32 Randomized Algorithm for OPIP Special case: b = 1 n unweighted p = 1 n single copy per item c = u n uniform capacities Algorithm: For each item j pick a random priority r(j) U[0, 1] Upon arrival of a constraint: Construct u subsets S i1,...,s iu Choose a ij subsets at random for item j For each subset S il reject all items but the one with highest priority Advantages: Relatively simple Oblivious decisions Shrinking Maxima, Decreasing Costs APPROX / 15

33 Intuition for a Weaker Result Reduction to unit capacities: Constraint i becomes u constraints Randomly choose a ij constraints j X a ij x j c i = j 8 X x k 1 k S X i1 x k 1 >< k S i2. X x >: k 1 k S iu Shrinking Maxima, Decreasing Costs APPROX / 15

34 Intuition for a Weaker Result Reduction to unit capacities: Constraint i becomes u constraints Randomly choose a ij constraints j Neighborhood of item j: N(j) = {k j : S il s.t. j, k S il } X a ij x j c i = j 8 X x k 1 k S X i1 x k 1 >< k S i2. X x >: k 1 k S iu Shrinking Maxima, Decreasing Costs APPROX / 15

35 Intuition for a Weaker Result Reduction to unit capacities: Constraint i becomes u constraints Randomly choose a ij constraints j Neighborhood of item j: N(j) = {k j : S il s.t. j, k S il } Claim: E[N(j)] < ρc X a ij x j c i = j 8 X x k 1 k S X i1 x k 1 >< k S i2. X x >: k 1 k S iu # of potential neighbors of j in constraint i is P k j a ik On average, each copy of j meets 1 of potential neighbors u E[N(j)] = P a ij P i u k j a ik < RC u = ρc Shrinking Maxima, Decreasing Costs APPROX / 15

36 Intuition for a Weaker Result Reduction to unit capacities: Constraint i becomes u constraints Randomly choose a ij constraints j Neighborhood of item j: N(j) = {k j : S il s.t. j, k S il } Claim: E[N(j)] < ρc X a ij x j c i = j 8 X x k 1 k S X i1 x k 1 >< k S i2. X x >: k 1 k S iu # of potential neighbors of j in constraint i is P k j a ik On average, each copy of j meets 1 of potential neighbors u E[N(j)] = P a ij P i u k j a ik < RC u = ρc Claim: Pr[j alg] 1 2ρC Markov Shrinking Maxima, Decreasing Costs APPROX / 15

37 Intuition for a Weaker Result Reduction to unit capacities: Constraint i becomes u constraints Randomly choose a ij constraints j Neighborhood of item j: N(j) = {k j : S il s.t. j, k S il } Claim: E[N(j)] < ρc X a ij x j c i = j 8 X x k 1 k S X i1 x k 1 >< k S i2. X x >: k 1 k S iu # of potential neighbors of j in constraint i is P k j a ik On average, each copy of j meets 1 of potential neighbors u E[N(j)] = P a ij P i u k j a ik < RC u = ρc Claim: Pr[j alg] 1 2ρC Markov alg is 2ρC-competitive: E[ alg ] j Pr[j alg] n 2ρC opt 2ρC Shrinking Maxima, Decreasing Costs APPROX / 15

38 Dual: Online Team Formation (OTF) PIP: max n j=1 b jx j s.t. n j=1 a ijx j c i x j p j x j N i j j Dual: min m i=1 c iy i + n j=1 p jz j m s.t. i=1 a ijy i + z j b j y i N z j N j i j Shrinking Maxima, Decreasing Costs APPROX / 15

39 Dual: Online Team Formation (OTF) PIP: max n j=1 b jx j s.t. n j=1 a ijx j c i x j p j x j N i j j Dual: min m i=1 c iy i + n j=1 p jz j m s.t. i=1 a ijy i + z j b j y i N z j N j i j Multi-Covering with Penalties: b j covering requirement of element j p j penalty of element j c i cost of set i a ij coverage of element j by set i Shrinking Maxima, Decreasing Costs APPROX / 15

40 Online Team Formation Motivation Special case: A {0, 1} n m, b = 1 n Skills: min s.t. We are embarking on a new project n skills are required Can obtain skill j by outsourcing for cost p j Candidates: Arrive online Candidate i has several skills and cost c i Interview dilemma: to hire or not to hire? A candidate not hired is gone! Goal: minimize project cost m i=1 c iy i + n j=1 p jz j n i=1 a ijy i + z j 1 y i {0, 1} z j {0, 1} j i j Shrinking Maxima, Decreasing Costs APPROX / 15

41 OTF Related Work Online Set Cover: [Alon Awerbuch Azar Buchbinder Naor 09] Items arrive online Cover grows with time Different model! Shrinking Maxima, Decreasing Costs APPROX / 15

42 OTF Related Work Online Set Cover: [Alon Awerbuch Azar Buchbinder Naor 09] Items arrive online Cover grows with time Different model! Secretary Problem: [Gilbert Mosteller 66], [Freeman 83] Similar to OTF: Decision about a candidate must be taken immediately upon arrival Candidates arrive in random order Goal is to pick best candidate Number of candidates is known Shrinking Maxima, Decreasing Costs APPROX / 15

43 OTF Results Lazy Approach: Do absolutely nothing! highest cost of covering j γ-competitive, where γ = max j lowest cost of covering j γ = ρ, if p = 1 m Shrinking Maxima, Decreasing Costs APPROX / 15

44 OTF Results Lazy Approach: Do absolutely nothing! highest cost of covering j γ-competitive, where γ = max j lowest cost of covering j γ = ρ, if p = 1 m Threshold Approach: Hire candidates that are γ-cost effective w.r.t. residual requirements O( γ)-competitive deterministic algorithm Requires prior knowledge of γ Shrinking Maxima, Decreasing Costs APPROX / 15

45 OTF Results Lazy Approach: Do absolutely nothing! highest cost of covering j γ-competitive, where γ = max j lowest cost of covering j γ = ρ, if p = 1 m Threshold Approach: Hire candidates that are γ-cost effective w.r.t. residual requirements O( γ)-competitive deterministic algorithm Requires prior knowledge of γ Proactive Approach: We are allowed to fire (but is not allowed to rehire) Ω( γ) randomized lower bound Applies even if γ is known Shrinking Maxima, Decreasing Costs APPROX / 15

46 OTF Threshold Algorithm Special Case: A {0, 1} n m b = 1 n, p = 1 n c = 1 m Shrinking Maxima, Decreasing Costs APPROX / 15

47 OTF Threshold Algorithm Special Case: A {0, 1} n m b = 1 n, p = 1 n c = 1 m γ = maximum number of skills per candidate Shrinking Maxima, Decreasing Costs APPROX / 15

48 OTF Threshold Algorithm Special Case: A {0, 1} n m b = 1 n, p = 1 n c = 1 m γ = maximum number of skills per candidate Threshold Algorithm: Hire arriving candidate if it covers γ uncovered skills Shrinking Maxima, Decreasing Costs APPROX / 15

49 OTF Threshold Algorithm Special Case: A {0, 1} n m b = 1 n, p = 1 n c = 1 m γ = maximum number of skills per candidate Threshold Algorithm: Hire arriving candidate if it covers γ uncovered skills Analysis : Covered skill: alg pays at most 1 γ, opt pays at least 1 γ Uncovered skill: alg pays at most 1, opt pays at least 1 γ Shrinking Maxima, Decreasing Costs APPROX / 15

50 OTF: Randomized Lower Bound There are p 2 skills, one for each point in Z p Z p Input Sequence: p 2 candidates, one for each line in Z p Z p Candidate i has skills that are contained in L i Shrinking Maxima, Decreasing Costs APPROX / 15

51 OTF: Randomized Lower Bound There are p 2 skills, one for each point in Z p Z p Input Sequence: p 2 candidates, one for each line in Z p Z p Candidate i has skills that are contained in L i Let h be the number of hired candidates: If E[h] [ p 2, p2 2 ]: stop = E[ alg ] = Ω(p 2 ), opt = O(p) Shrinking Maxima, Decreasing Costs APPROX / 15

52 OTF: Randomized Lower Bound There are p 2 skills, one for each point in Z p Z p Input Sequence: p 2 candidates, one for each line in Z p Z p Candidate i has skills that are contained in L i Let h be the number of hired candidates: If E[h] [ p 2, p2 2 ]: stop = E[ alg ] = Ω(p 2 ), opt = O(p) If E[h] [ p 2, p2 2 ]: Choose random L i, Introduce a new candidate: Z p Z p \ L i = E[ alg ] = Ω(p), opt = O(1) Shrinking Maxima, Decreasing Costs APPROX / 15

53 OTF: Randomized Lower Bound There are p 2 skills, one for each point in Z p Z p Input Sequence: p 2 candidates, one for each line in Z p Z p Candidate i has skills that are contained in L i Let h be the number of hired candidates: If E[h] [ p 2, p2 2 ]: stop = E[ alg ] = Ω(p 2 ), opt = O(p) If E[h] [ p 2, p2 2 ]: Choose random L i, Introduce a new candidate: Z p Z p \ L i = E[ alg ] = Ω(p), opt = O(1) Lower Bound: Ω(p) = Ω( γ) Shrinking Maxima, Decreasing Costs APPROX / 15

54 Conclusion Summary: O(C ρ)-competitive algorithm for OPIP O( γ)-competitive deterministic algorithm for OTF Ω( γ) randomized lower bound for OTF Shrinking Maxima, Decreasing Costs APPROX / 15

55 Conclusion Summary: O(C ρ)-competitive algorithm for OPIP O( γ)-competitive deterministic algorithm for OTF Ω( γ) randomized lower bound for OTF Open Problems: OPIP: lower bound for uniform non-unit capacity Shrinking Maxima, Decreasing Costs APPROX / 15

56 Conclusion Summary: O(C ρ)-competitive algorithm for OPIP O( γ)-competitive deterministic algorithm for OTF Ω( γ) randomized lower bound for OTF Open Problems: OPIP: lower bound for uniform non-unit capacity OTF: O( γ)-competitive algorithm when γ is unknown Algorithm must be allowed to fire candidates Shrinking Maxima, Decreasing Costs APPROX / 15

57 Conclusion Summary: O(C ρ)-competitive algorithm for OPIP O( γ)-competitive deterministic algorithm for OTF Ω( γ) randomized lower bound for OTF Open Problems: OPIP: lower bound for uniform non-unit capacity OTF: O( γ)-competitive algorithm when γ is unknown Algorithm must be allowed to fire candidates Shrinking Maxima, Decreasing Costs APPROX / 15