Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental Zvi Lotker Ben-Gurion University Boaz Patt-Shamir Dror Rawitz Tel Aviv University

Rent or Buy Dilemma the sleeping Baby Problem: You finally managed to put baby to sleep Baby will wake up at some unknown time Should parent stay awake or go to sleep? – Going to sleep incurs some fixed effort – Staying awake incurs an effort per time unit

Rent or Buy Dilemma Question: Competitive Analysis: How do we measure the quality of our solution Competitive Analysis: Worst case analysis: Baby will try to make parent’s life hard Compare our solution to best possible solution

Rent or Buy Classical Ski Rental: – Vacation at ski resort – End of vacation is unknown – Cost of skis is €B, rent is €1/day – Should we rent or buy the skis? – When should we buy? Optimal offline cost:

Optimal Online Strategies: 2-competitive deterministic strategy [Karlin et al. 88] -competitive randomized strategy [Karlin et al. 94] What if the second option is not const?

General two slope 1+(1-r)-competitive deterministic strategy Assume that we can chose: To rent or to buy hose But even when we by the hose we need to pay the property tax 1+(1-r)-competitive deterministic strategy -competitive randomized strategy

Deterministic strategy If the end of the game is at t<1 Opt cost t If the end of the game is at t>1 Opt cost r·t+(1-r) Assume the game end at the time online move to the second option In this case online cost is t+1-r So the competitive is

Ski rental with two general options rate due to investment in buying slope 2 third term is due to being at slope 2 Let pi(t) be the probability that the algorithm is using slope i=1,2 The expected rate in which the algorithm spends money is

How we find c At time t>1,optimal strategy spends money at rate a Assume that the online stop buying

Interpretation of the algorithm We pick an random time according to P2 If this number is bigger than 1 we do not move to the second option If this number is less then 1 we buy the second option at that time. Example assume that r=0.3, c=1.34683

Optimality of the algorithm We use Yao’s Lemma We assume that the game end at time t with the prob The optimal expected cost is

Optimality of the algorithm Assume that Ax is a deterministic algorithm that end x them the expected cost of Ax is 1, for 0<x<1 Therefore the competitive is

Rent, Lease or Buy Housing costs game: Extended Ski Rental: Price of house/apartment increases closer to city center Transportation rates decrease closer to city center Extended Ski Rental: Mixed rent and buy options Pure buy or pure rent may not exist

Multislope Ski Rental Problem Definition: End time is unknown Several states/slopes Slope i: bi+ri·t bi+1>bi for all i ri+1<ri for all i End time is unknown Which slope should we buy? When?

Multislope Ski Rental Online Buying Costs: Say we are in slope i, how much do we pay for slope j Additive Model: bj−bi Non-Additive Model: bij “From scratch”: bj

Offline Strategy Competitive Analysis: The online strategy is compared to the offline strategy Offline Strategy: Buy a slope at time 0 according to end time

Previous Results & Applications Online Capital Investments: Deterministic 6.83-competitive strategy (slopes may arrive over time) [Azar et al. 99] Deterministic lower bound 3.618 Randomized 2.88-competitive strategy [Damaschke 03] Rerouting in ATM networks: 4-competitive strategy (slopes may be concave) [BCN 00]

Previous Results & Energy Saving Slopes are hibernation modes Deterministic 2-competitive strategy for additive model [IGS 02] Algorithm that computes best deterministic strategy for non-additive model [AIS 04]

Our Results Additive Model: Main Result: Randomized e/(e−1) competitive strategy (e−rk/r0)/(e−1) when rk >0 Decomposition into k classical ski rental instances Strategy is combination of k strategies Main Result: Algorithm that computes best randomized strategy

Profiles Costs Expected rent at time t: Expected total rental cost at time t: Expected buying cost at time t: Total: Expected cost at time t:

This can by approximate by lp diff. equations This can by approximate by lp The problem we want to solve is

So we ran lp aproximation And this is what we got

Strategies & Profiles Randomized Startegy: Randomized Profile: Plan: Additive model: we go from slope i to slope i + 1 When do we move to the next slope? Probability distribution over deterministic strategies Randomized Profile: pi(t) – probability of being in slope i at time t Σipi(t) = 1 for all t≥0 Plan: Every strategy induces a profile Find best profile Construct strategy

Optimal Profiles Chain of Transformations: Optimal profile Continuous optimal profile Continuous in t Prudent optimal profile Only one or two consecutive active slopes Tight optimal profile Moves to the next slope as soon as possible

Prudent Profiles Continuous to Prudent: for all t two consecutive slopes are determined Buying cost is preserved ⇛ Rent may only decrease ⇛ Continuity is preserved

Prudent to Tight Tight: Buy next slope as soon as possible ⇛ Rent may only decrease Theorem: There exists a tight optimal profile

Computing Tight Profile Computing tight profile for a given c: Solve diff. equations: For tight profiles: We assume that we know c and then we solve all those diff. equations If we solve all of them before time 1 we can make c small If not we have to make c bigger

Algorithm Computing a Strategy: Let p be a tight profile X~U(0, 1) Move from slope i to slope i + 1 when Theorem: Randomized (c + ε)-competitive strategy can be found in O(k log 1/ε), where c is the best possible ratio

3-Slope Examples:

An 1.581 Competitive Algorithm Assume rk=0 Let b’0=0; r’i0=ri-1-ri; b’i1=bi-bi-1; r’i1=0 Let Opti(t) be the offline alg for ski prob {(b0i, r0i), (b1i, r1i)} Lemma Opt(t)=Σ Opti(t) Proof:

An 1.581 Competitive Algorithm We solve each problem separate Let P0i(t), P1i(t) be the solution. We define the profile for the multislope: Pi(t)= P1i(t)- P1i+1(t) for i=1,…,k-1 P0(t)= P01(t), Pk(t)= P1k(t)

An 1.581 Competitive Algorithm We solve each problem separate Let P0i(t), P1i(t) be the solution. Lemma P1i-1(t)≥P1i(t) Proof P1i(t)=Exp[b1i/r0i] now b1i/r0i>b1i-1/r0i-1 It is clear that the sum of all prob is 1.

An 1.581 Competitive Algorithm Given P one can obtain an online strategy whose profile is P. let U~U[0,1] we move from state i to state i + 1 when U=P1i(t) for every state i

An 1.581 Competitive Algorithm Theorem The competitive ratio of P is: e/(e−1) Proof: expected cost to the combined strategy is the sum of the costs to the two-slope strategies buying cost is ΣBPi(t) Ranting cost is ΣrPi(t) By the fact that each of the strategies is e/(e-1) competitive the lemma follows.

Open Problems: Compute best randomized strategy for non-additive model What is the get LP for homogeneous differential equation.

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