VIEWS: 1 PAGES: 45 POSTED ON: 3/23/2011 Public Domain
Truthful Approximation Mechanisms for Scheduling Selfish Related Machines Motti Sorani, Nir Andelman & Yossi Azar Tel-Aviv University The Classic Scheduling Problem • Scheduling jobs on uniformly related machines – n Jobs: ( p1 , p2 ,.., pn ) – m machines with different speeds: s1 , s2 ,..., sm – The objective: minimize the maximum completion time over all machines (Makespan) • Known to be NP-Complete • Several approximation-algorithms Scheduling Jobs on Selfish Related Machines • One-Parameter problem • Machines are owned by rational selfish agents Scheduling Jobs on Selfish Related Machines • The machine’s speed si is known to its owner only. The secret: the cost per unit of work ti 1 / si t1 2 / 3 t2 1 t3 1 Scheduling Jobs on Selfish Related Machines • Job sizes are common knowledge • The system wants to execute the jobs while minimizing the makespan t1 2 / 3 t2 1 t3 1 Scheduling Jobs on Selfish Related Machines • Bidding b1 1 / 2 b2 1 b3 1 t1 2 / 3 t2 1 t3 1 Scheduling Jobs on Selfish Related Machines • The cost of machine i is ti wi where wi is the total amount of work assigned to it. • The machines get paid. The goal of each machine is to maximize its profit. b1 1 / 2 b2 1 b3 1 t1 2 / 3 t2 1 t3 1 Watching the Game The jobs: 12,10, 7, 5, 4, 4, 3 t1 2 / 3 t2 1 t3 1 Watching the Game The jobs: 12,10, 7, 5, 4, 4, 3 First Phase: Bidding b1 1 / 2 b2 1 b3 1 t1 2 / 3 t2 1 t3 1 Watching the Game The jobs: 12,10, 7, 5, 4, 4, 3 Second Phase: The system allocates the jobs to the machines according to their declared bids, and simultaneously delivers the payments Makespan=12 5 3 10 4 12 7 4 b1 1 / 2 b2 1 b3 1 16 15 13 t1 2 / 3 t2 1 t3 1 Truthful Mechanism Design M=(A,P) A : Allocation Algorithm P : Payment t1 t2 t3 Truthful Mechanism Design M=(A,P) A : Allocation Algorithm P : Payment b1 t1 b2 t2 b3 t3 t1 t2 t3 Mechanism Design • The idea: Overcome selfishness by payments • Mechanism M=(A,P) • Strategy for agent i : bi • The outcome of the algorithm is o(b) • The work allocated to agent i : wi (o(b)) • The payment to agent i : Pi (b) • The profit of agent i : profit i Pi (b) ti wi (o(b)) Observation: Paying each agent its cost is not truthful Truthful Mechanism • Dominant strategy for agent i : profit i (bi , bi ) profit i (bi , b'i ) for all bi , b'i • Truthfulness: truth-telling ( bi ti ) is a dominant strategy for each agent • VCG is not applicable as the objective is not utilitarian (maximize “social welfare”) Our goal: Design a Truthful Mechanism M=(A,P) which approximates the Minimum Makespan Truthful Mechanism • Consider the work assigned to agent i as a single-variable function of bi • Work-curve wi (bi , bi ) ti bi Truthfulness <=> Monotone Algorithm Truthful Mechanism Theorem [Archer and Tardos]: A mechanism is truthful and admits a voluntary participation iff (a) the work-curve for each agent is decreasing, (b) 0 wi (bi , u )du and the payments in this case should be Pi (bi , bi ) bi wi (bi , bi ) wi (bi , u)du bi Monotone Algorithms • Truthful Mechanism: – Monotone Algorithm – Payment scheme Pi (bi , bi ) bi wi (bi , bi ) wi (bi , u)du bi • The work-curve wi (bi , bi ) profit cost ti bi Overbidding wi (bi , bi ) less profit less payment slower bi faster ti x Underbidding wi (bi , bi ) loss bi x ti Previous Results - Approximation • Gonzalez et al: 2-approximation LPT greedy assignment • Horowitz and Sahni: FPTAS for constant number of machines • Hochbaum and Shmoys: PTAS for arbitrary number of machines All these algorithms are not monotone Previous Results – Mechanism Design • Monotone Algorithm (not polytime) [Archer & Tardos]: – optimal solution – satisfies voluntary participation Among the optimal allocations of jobs, select the one in which the work-vector ( w1 , w2 ,.., wm ) is lexicographically minimum. Previous Results – Mechanism Design Scenario: gradual slowdown Slowing down Previous Results – Mechanism Design • Classic approximation algorithms are not monotone. • Archer and Tardos: randomized truthful 3-approximation mechanism (truthful in expectation) • Auletta et al: deterministic truthful (4+ε)-approximation mechanism for any fixed number of machines Notions of Truthfulness • Truthfulness in expectation: bidding truthfully always maximizes the agent's expected profit • Universal truthfulness bidding truthfully always maximizes an agent's profit, no matter what the other agents bid, and no matter what are the outcomes of the mechanism's random coin flips Our Results • Deterministic 5-approximation truthful mechanism for arbitrary number of machines • Deterministic truthful (F)PTAS for any fixed number of machines We now show a simplified version for arbitrary number of machines which achieves a 12-approximation truthful mechanism. Valid Fractional Assignment • Given a threshold T, treat the machines as bins of size T/bi • Fractional Assignment – Partition each job to pieces, assign the pieces to the bins • Valid Fractional Assignment – Each bin is large enough to contain all pieces assigned to it – For every piece assigned to a bin, the bin is capable of containing the entire job (which the piece belongs to) • T f – The smallest threshold for which a valid fractional assignment exists. Valid Fractional Assignment • Example – Jobs: 7,5,4,3,3,2 – Bids: 1/5, 1/4, 1/3 – Threshold = 2 3 2 5 4 3 7 5 3 bin size: 10 8 6 Valid Fractional Assignment • T f can be calculated in a greedy manner j p k T max min max{ bi p j , k 1 f i } 1 b j i l 1 l • Tf is a lower-bound to Opt Monotonicity of Tf • Observation: T f (bi , bi ) T f (bi , bi ) for all 1 and i • T f behaves in a “monotone manner” – For any machine i which is not the fastest (i>1) T f (bi , bi ) T f (bi , ) 2T f (bi , bi ) for all bi Truthful Mechanism for arbitrary number of Machines • Guidelines of algorithm Monotone-RF Init: – Round the bids to the closest power of 4 – Sort the jobs in non-increasing order fractional: – Calculate a valid fractional assignment and an appropriate threshold Tf rounding: – Assign jobs (using the rounded bids) in non- increasing order of size, from the fastest to the slowest (breaking ties by external ID) • The first machine – until a threshold of 2Tf is exceeded • Rest of the machines – until a threshold Tf is exceeded – Return the assignment Truthful Mechanism for arbitrary number of Machines 2T f Tf fastest slowest … Monotonicity of Monotone-RF • Intuition: Assigning jobs according the rounded bids forces non-increasing work-curves • From now on we assume the bids are equal to the real speeds. • We shall show : Slowing down => Less/Equal Amount of work Why Rounding the Bids Helps? • T f behaves in a “monotone manner” – For any machine i which is not the fastest (i>1) T f (bi , bi ) T f (bi , ) 2T f (bi , bi ) for all bi • Say the rounded bid is multiplied by 4 d i rounded bi ; vi 1 / d i Total work is 2T f At most v f Total 2(2T f )( i ) T 4 work is T vi f At least T f vi vi vi / 4 Scenarios of Slowdowns • The unique fastest behaves differently: Rounding is not enough T f (b1 , b1 ) T f (b1 , ) 4T f (b1 , b1 ) for all b1 • The bad scenario: The fastest machine slows down one step (the rounded bid is multiplied by 4) and some other machine becomes the fastest Scenarios of Slowdowns • Solution: Double the threshold for the bin of the unique fastest machine (2T f ) • When it slows down one step ,two cases: – Remains the unique fastest: • Bin size can not increase • Jobs are allocated in the same order – No longer the fastest: losing the doubled threshold balances the possible increase of the threshold. Analysis for Partially-Full and Empty Machines • So far we considered full machines only The Red Machine slows down new T f Tf Monotonicity by Gradual Slowdown • Monotone-RF is monotone. Hence a Mechanism based on Monotone-RF and payment scheme Pi (bi , bi ) bi wi (bi , bi ) wi (bi , u)du bi is truthful Truthfulness - Remark • The work-curve Approximation Analysis • The first bin capacity: 3T f • T f is a lower-bound to Opt. • Speeds were rounded to powers of 4 • A Total of 12-approximation Guidelines for 5-Approximation • Prefer the fastest machine already in the rounding phase. • Make sure the first bin is at least 4 times the second one – For any machine i which is not the fastest (i>1) 5 f T (bi , bi ) T (bi , ) T (bi , bi ) for all bi f f 4 • Speeds can be rounded to powers of 2.5 • A Total of 5-approximation Truthful PTAS-Mechanism for Any Fixed Number of Machines n p i 1 j 1 1 2 2 [ 2 , 2] 2m m Exact Minimum-Lexicographically Solution Truthfulness of Monotone-PTAS • Job sizes were generated independently from the bids • The optimal Min. Lex. Solution is monotone Approximation Analysis • Running Time is linear • Assume we do not use machines slower than smax m • The chunks add multiplicative overhead of (1+ε) • The assumption above adds another multiplicative overhead of (1+ε) • T (1 ) Opt (1 3 )Opt , for any 0.5 2 Guidelines for the FPTAS • Uses any c-approximation algorithm as a black box, generates a c(1+ε)-approximation • Rounds the bids to powers of (1+ε) • Calculate all possible (sorted) assignments made by the black box • Try all assignments on the given rounded bids- vector. Pick the one with minimal makespan, or if more than one exists, the one which is lexicographically maximum. Conclusions and Open Problems • We have shown – Deterministic 5-approximation truthful mechanism for assigning jobs on related machines – A (F)PTAS truthful mechanism for any fixed number of machines • Is there a PTAS truthful mechanism for arbitrary number of Machines?