TOLA 2014 Christine Markarian July 7, 2014 Joint work with: - PowerPoint PPT Presentation

The Price of Leasing Online TOLA 2014 Christine Markarian July 7, 2014 Joint work with: Sebastian Abshoff Peter Kling Friedhelm Meyer auf der Heide 1 Christine Markarian

Outline 2 Christine Markarian

Parking Permit Problem sunny day walk rainy day drive [Meyerson - FOCS 2003] 3 Christine Markarian

Parking Permit Problem  adversary gives sunny or rainy on each day  K permit lease types ( Ex. daily, weekly, monthly, yearly)  yearly permit is the most expensive but cheapest per day Which permits do I buy & when in order to provide every rainy day with a permit? Online algorithm quality of online algorithm → competitive factor α • adversary reveals in each step part of overall input Optimal Offline 𝐝𝐩𝐭𝐮 𝐩𝐠 𝐏𝐨𝐦𝐣𝐨𝐟 𝐛𝐦𝐡𝐩𝐬𝐣𝐮𝐢𝐧 • 𝛽 = max knows 𝐝𝐩𝐭𝐮 𝐩𝐠 𝐏𝐪𝐮𝐣𝐧𝐛𝐦 𝐏𝐠𝐠𝐦𝐣𝐨𝐟 𝐛𝐦𝐡𝐩𝐬𝐣𝐮𝐢𝐧 over all input instances the future in advance [Meyerson - FOCS 2003] 4 Christine Markarian

Parking Permit Problem Lower bounds Upper bounds Ω(𝐿 ) deterministic O 𝐿 deterministic Ω(log 𝐿 ) randomized O log 𝐿 randomized Deterministic algorithm For each rainy day, buy a 1-day permit, until there is some (𝑙 ∈ 𝐿) -interval where the optimum offline solution for the sequence of days seen so far, would buy a 𝑙 -day permit. In this case, also buy a 𝑙 -day permit. Randomized algorithm Compute an 𝑃 log 𝐿 -competitive fractional solution and then convert it into a randomized integer solution which maintains the 𝑃 log 𝐿 -competitive factor. [Meyerson - FOCS 2003] 5 Christine Markarian

Outline 6 Christine Markarian

Infrastructure Leasing Problems Provider 1 Leasing Company Client 1 Provider 2 Client 2 Client 3 Provider 3 . . . . . . Long lease or short lease ….. ? 7 Christine Markarian

Infrastructure Leasing Problems  Almost any online infrastructure problem can be considered with a leasing aspect….  Anthony & Gupta generalized the Parking Permit Problem • Facility Leasing • Steiner Tree Leasing • Set Cover Leasing & gave offline algorithms to the problems… [Anthony et al.- IPCO 2007] 8 Christine Markarian

Outline 9 Christine Markarian

Set Cover Leasing Online Set cover Set Cover Leasing • U = { e 1 , e 2 ,.…., e n } • U = { e 1 , e 2 ,.…., e n } • family F = { S 1 , S 2 ,.…., S m } of subsets of • family F = { S 1 , S 2 ,.…., S m } of subsets of U and a cost associated with each U. Each set in F can be leased for K subset different periods of time such that leasing a set S for a period k : • incurs a cost c kS • allows S to cover its elements for the next l k time steps • an element e ∈ U arrives • an element e ∈ U arrives -- choose sets from F to cover each arriving -- lease sets from F to cover each arriving element e ∈ U & minimize cost of sets -- element e ∈ U & minimize cost of sets -- generalizes Online Set Cover ( K = 1) - one infinite lease - 10 Christine Markarian

Set Cover Leasing • e 1 arrives at time t e 1 ∈ { S 3 , S 5 , S 8 } • e 1 t’ t’’ t T S 3 S 3 S 5 . . . 11 Christine Markarian

Set Cover Leasing Ex. servers/clients in a computer network a client arrives in each step Which servers shall I install in order to serve each arriving client while minimizing cost ?? network manager Once a server is installed, it serves its clients forever without additional costs… [Online set cover] If servers are leased instead & can serve their clients only during the time they are leased… [Set cover leasing] 12 Christine Markarian

Set Cover Leasing Lower bounds -none -related problems log 𝑜 -Online Metric Facility Location: 𝛻( log log 𝑜 ) [ICALP 2003] 𝑚𝑝𝑕 𝑜 log 𝑛 -Online Set Cover: 𝛻( 𝑚𝑝𝑕 𝑚𝑝𝑕 𝑜+log log 𝑛 ) [STOC 2003] Upper bounds -Online Metric Facility Leasing: O 𝐿 𝑚𝑝𝑕 𝑜 [IPCO 2008] -An algorithm for Online Facility Leasing: O 𝑚_𝑛𝑏𝑦 𝑚𝑝𝑕 𝑚_𝑛𝑏𝑦 [SIROCCO 12] -Randomized Online Algorithms for Set Cover Leasing Problems: O log(𝑛𝐿) 𝑚𝑝𝑕 𝑜 [submitted to WAOA] 13 Christine Markarian

Set Cover Leasing Algorithm {Set Cover Leasing} Maintain a fraction 𝑔  Given : F = {S 1 , S 2 ,.…., S m }, K 𝑇𝑙𝑢 for each set ( S , k , t ) leases, U = { e 1 , e 2 ,.…., e n } • set to 0 initially ( S , k , T ): S ∈ 𝐺 , lease k , interval T  • non-decreasing throughout algorithm ( j, t ): i ∈ 𝑉 , arrives at time t   ( S , k , T ) is a candidate of ( j, t, ), if j Maintain for each set ( S , k , t ) ∈ 𝑇 & t ∈ 𝑈 2 log(𝑜 + 1) independent random variables 𝑌 (𝑇𝑙𝑢)(𝑟) in 0, 1 • 𝑹 𝒌 is the set of candidates of 𝑘  • Let 𝜈 𝑇𝑙𝑢 = min 𝑌 (𝑇𝑙𝑢)(𝑟) , 1 ≤ 𝑟 ≤ 2 log(𝑜 + 1) ( j, t ) arrives, i. (fractional) If 𝑇,𝑙,𝑈 ∈𝑅 𝑘 𝑔 𝑇𝑙𝑢 < 1 , do the following increment 𝑥ℎ𝑗𝑚𝑓 𝑇,𝑙,𝑈 ∈𝑅 𝑘 𝑔 𝑇𝑙𝑢 < 1 ; 𝑇𝑙𝑢 ∙ 1 + 1 1 𝑔 𝑇𝑙𝑢 = 𝑔 + 𝑑 𝑙𝑇 𝑅 𝑘 ∙ 𝑑 𝑙𝑇 ii. (integer) Lease ( S , k , T ) ∈ 𝑅 𝑘 with 𝑔 𝑇𝑙𝑢 > 𝜈 𝑇𝑙𝑢 𝑷 𝐦𝐩𝐡 (𝒆𝑳) 𝐦𝐩𝐡 𝐨 − 𝒅𝒑𝒏𝒒𝒇𝒖𝒋𝒖𝒋𝒘𝒇 iii. If ( j, t ) is not covered by some set in 𝑅 𝑘 (i) 𝑔𝑠𝑏𝑑𝑢𝑗𝑝𝑜𝑏𝑚 ≤ 𝑃 log(𝑒𝐿 ) ∙ 𝑃𝑞𝑢 Lease the cheapest ( S , k , T ) ∈ 𝑅 𝑘 (ii) 𝑠𝑏𝑜𝑒𝑝𝑛𝑗𝑨𝑓𝑒 𝑗𝑜𝑢𝑓𝑕𝑓𝑠 ≤ 𝑃 log 𝑜 ∙ 𝑔𝑠𝑏𝑑𝑢𝑗𝑝𝑜𝑏𝑚 (iii) 𝑡𝑢𝑓𝑞 𝑗𝑗𝑗 𝑏𝑒𝑒𝑡 𝑏𝑜 𝑓𝑦𝑞𝑓𝑑𝑢𝑓𝑒 𝑑𝑝𝑡𝑢 𝑝𝑔 𝑃𝑞𝑢/𝑜 14 Christine Markarian

Set Cover Leasing Algorithm {i-cover} ( j, t ) arrives. 𝑷 𝐦𝐩𝐡 (𝒆𝑳) 𝐦𝐩𝐡 𝐨 − 𝒅𝒑𝒏𝒒𝒇𝒖𝒋𝒖𝒋𝒘𝒇 (fractional) If 𝑇,𝑙,𝑈 ∈𝑅 𝑘 𝑔 𝑇𝑙𝑢 < 1 , do the i. following increment (i) 𝑔𝑠𝑏𝑑𝑢𝑗𝑝𝑜𝑏𝑚 ≤ 𝑃 log(𝑒𝐿 ) ∙ 𝑃𝑞𝑢 𝑥ℎ𝑗𝑚𝑓 𝑇,𝑙,𝑈 ∈𝑅 𝑘 𝑔 𝑇𝑙𝑢 < 1 ; (ii) 𝑠𝑏𝑜𝑒𝑝𝑛𝑗𝑨𝑓𝑒 𝑗𝑜𝑢𝑓𝑕𝑓𝑠 ≤ 𝑃 log 𝑜 ∙ 𝑔𝑠𝑏𝑑𝑢𝑗𝑝𝑜𝑏𝑚 𝑇𝑙𝑢 ∙ 1 + 1 1 𝑔 𝑇𝑙𝑢 = 𝑔 + 𝑑 𝑙𝑇 𝑅 𝑘 ∙ 𝑑 𝑙𝑇 (iii) 𝑡𝑢𝑓𝑞 𝑗𝑗𝑗 𝑏𝑒𝑒𝑡 𝑏𝑜 𝑓𝑦𝑞𝑓𝑑𝑢𝑓𝑒 𝑑𝑝𝑡𝑢 𝑝𝑔 𝑃𝑞𝑢/𝑜 ii. (integer) Lease ( S , k , T ) ∈ 𝑅 𝑘 with 𝑔 𝑇𝑙𝑢 > 𝜈 𝑇𝑙𝑢 iii. If ( j, t ) is not covered by some set in 𝑅 𝑘 Proof: (i) Lease the cheapest ( S , k , T ) ∈ 𝑅 𝑘 - an 𝑗𝑜𝑑𝑠𝑓𝑛𝑓𝑜𝑢 adds at most 2 to the 𝑔𝑠𝑏𝑑𝑢𝑗𝑝𝑜𝑏𝑚 𝑑𝑝𝑡𝑢 𝑑 𝑇 ∙ 𝑔 1 𝑇𝑙𝑢 + ∙= 𝑔 𝑇𝑙𝑢 + 1 ≤ 2 𝑑 𝑇 𝑅 ∙ 𝑑 𝑇 𝑇,𝑙,𝑈 ∈𝑅 𝑇,𝑙,𝑈 ∈𝑅 - the total number of 𝑗𝑜𝑑𝑠𝑓𝑛𝑓𝑜𝑢𝑡 in the algorithm is 𝑃 log(𝑒𝐿 ) ∙ 𝑃𝑞𝑢 At any time the algorithm decides to make an 𝑗𝑜𝑑𝑠𝑓𝑛𝑓𝑜𝑢 , ∃ 𝑇 𝑝𝑞𝑢 which is a candidate and • therefore increases its fraction 𝑔 𝑇 𝑝𝑞𝑢 𝑙𝑢 𝑇 𝑝𝑞𝑢 𝑙𝑢 > 1 → 𝑇,𝑙,𝑈 ∈𝑅 𝑔 • After 𝑃(𝑑 𝑇 ∙ log 𝑅 ) 𝑗𝑜𝑑𝑠𝑓𝑛𝑓𝑜𝑢𝑡, 𝑔 𝑇𝑙𝑢 > 1 Parking Permit 𝑅 ≤ 𝑒 ∙ 𝐿 [ Interval Model: Same sets same leases do not coincide] • Problem 15 Christine Markarian

Set Cover Leasing Proof: (ii) 𝑠𝑏𝑜𝑒𝑝𝑛𝑗𝑨𝑓𝑒 𝑗𝑜𝑢𝑓𝑕𝑓𝑠 ≤ 𝑃 log 𝑜 ∙ 𝑔𝑠𝑏𝑑𝑢𝑗𝑝𝑜𝑏𝑚 - Probability to lease a set is 𝑄𝑠( 𝑔 𝑇𝑙𝑢 > 𝜈 𝑇𝑙𝑢 ) - 𝜈 𝑇𝑙𝑢 = min 𝑌 (𝑇𝑙𝑢)(𝑟) , 1 ≤ 𝑟 ≤ 2 log(𝑜 + 1) Proof: (iii) 𝑡𝑢𝑓𝑞 𝑗𝑗𝑗 𝑏𝑒𝑒𝑡 𝑏𝑜 𝑓𝑦𝑞𝑓𝑑𝑢𝑓𝑒 𝑑𝑝𝑡𝑢 𝑝𝑔 𝑃𝑞𝑢/𝑜 - [Algorithm leases the cheapest ( S , k , T ) ∈ Q] 𝑑 𝑇 ≤ 𝑃𝑞𝑢 - Probability that an element is not covered [for a single 𝑟] is at most 𝑇𝑙𝑢 ≤ 𝑓 − 𝑇,𝑙,𝑈 ∈𝑅 𝑔 𝑇𝑙𝑢 ≤ 1/𝑓 1 − 𝑔 (𝑇𝑙𝑢)∈𝑅 - Probability that an element is not covered is at most 1/𝑜 2 1 - 𝑏𝑒𝑒𝑗𝑢𝑗𝑝𝑜𝑏𝑚 𝑓𝑦𝑞𝑓𝑑𝑢𝑓𝑒 𝑑𝑝𝑡𝑢 ≤ 𝑜 ∙ 𝑜 2 ∙ 𝑃𝑞𝑢 → 𝑃 𝑚𝑝𝑕 (𝑒𝐿) 𝑚𝑝𝑕 𝑜 − 𝑑𝑝𝑛𝑞𝑓𝑢𝑗𝑢𝑗𝑤𝑓 16 Christine Markarian

Outline 17 Christine Markarian

The Price of Leasing Online Lower bounds 𝑚𝑝𝑕 𝑜 log 𝑛 log 𝑜 Online Set Cover: 𝛻( Online Facility Location : 𝛻( 𝑚𝑝𝑕 𝑚𝑝𝑕 𝑜+log log 𝑛 ) log log 𝑜 ) + + Parking Permit Problem: Ω 𝐿 Parking Permit Problem: Ω 𝐿 ? ? ….. Leasing algorithms so far use techniques from non-leasing algorithms & Parking Permit Problem... Does leasing impose an inherent difficulty? What is the price we pay for leasing? 18 Christine Markarian

Thank you for your attention! Christine Markarian July 7, 2014 19 Christine Markarian