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Truthful Mechanism Design for Multi-Dimensional Scheduling via Cycle Monotonicity

Truthful Mechanism Design for Multi-Dimensional Scheduling via Cycle Monotonicity. Ron Lavi IE&M, The Technion. Chaitanya Swamy U. of Waterloo. and. Job scheduling. n tasks (“jobs”) to be assigned to m workers (“machines”) Each machine, i, needs p ij time units to complete job j.

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Truthful Mechanism Design for Multi-Dimensional Scheduling via Cycle Monotonicity

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  1. Truthful Mechanism Design for Multi-Dimensional Scheduling via Cycle Monotonicity Ron Lavi IE&M, The Technion Chaitanya Swamy U. of Waterloo and

  2. Job scheduling • n tasks (“jobs”) to be assigned to m workers (“machines”) • Each machine, i, needs pij time units to complete job j. • Our goal: to assign jobs to machines to complete all jobs as soon as possible. More formally: • Let Si denote the set of jobs assigned to machine i, and define the load of a machine: li = jSipij. • Our goal is then to minimize the maximal load (a.k.a the “makespan” of the schedule).

  3. Example Two machines, three jobs: Job p1j p2j 1 2 1 2 2 3 3 3 4 A possible assignment: J2 J3 J1 1 2

  4. Example Two machines, three jobs: Job p1j p2j 1 2 1 2 2 3 3 3 4 A possible assignment: l1 = 3 l2 = 4 J2 J3 J1 Makespan = 4 1 2

  5. Scheduling and Mechanism Design • The workers/machines are selfish entities, each one is acting to maximize her individual utility. • If job j is assigned to machine i, it will incur a cost pij for executing the job. • A machine may get a payment, Pi, and its total utility is: Pi - li • Question: design a truthful mechanism (in dominant strategies) that will reach a “close to optimal” makespan. • First raised by Nisan and Ronen (GEB, 2001). • Basic observation: makespan minimization is inherently different than welfare maximization, hence VCG performs poorly (obtains makespan of up to m times the optimum).

  6. Example Two machines, three jobs: Job p1j p2j 1 2 1 2 2 3 3 3 4 A possible assignment: J2 J3 J1 Max. Makespan = 4 Tot. Welfare = -3 - 3 -1 = -7 1 2

  7. Example Two machines, three jobs: Job p1j p2j 1 2 1 2 2 3 3 3 4 A possible assignment: J2 J3 J1 Max. Makespan = 5 Tot. Welfare = -2 - 3 -1 = -6 1 2

  8. Why is this question important? (1) • Significant to several disciplines: • Computer Science • Operations Research • Makespan minimization is similar to a Rawls’ max-min criteria -- gives a justification from social choice theory. • The implicit goal: assign tasks to workers in a fair manner (rather than in a socially efficient manner). • Can we do it via classic mechanism design?

  9. Why is this question important? (2) • The general status of mechanism design for multi-dimensional domains is still unclear. • What social choice functions can be implemented? • Few possibilities, few impossibilities, more questions than answers. • Scheduling is a multi-dimensional domain, and is becoming one of the important domains for which we need to determine the possibilities - impossibilities border.

  10. Current status (1) • Nisan and Ronen (1999): a lower bound of 2 for truthful deterministic approximations (regardless of computational issues). • But only give a m-approximation upper bound (VCG) -- the gap is very large. • Christodoulou, Koutsoupias, and Vidali (2007): an improved lower bound (about 2.4). • Mu’alem and Schapira (2007): a 2-(1/m) lower bound for randomized mechanisms and truthfulness in expectation. • No non-trivial truthful approximation (i.e. o(m)) is known!

  11. Current status (2) • Archer and Tardos (2001) study the special case of related machines: each machine has speed si,and pij = pj/si. • The optimum is implementable (but NP-hard). • Many truthful approximations suggested since. The current-best: a deterministic 3-approximation by Kovacs (2005). • Also, a truthful PTAS for a fixed number of machines, by Andelman, Azar, and Sorani (2004). • Note: this is a single-dimensional domain, thus it demonstrates again the contrast between single and multi dimensionality.

  12. A multi-dimensional special case • We study a special case of two fixed values: pij {Lj , Hj} • Values are fixed and known to the mechanism. • Still a multi-dimensional domain. • Generalizes the classic “restricted machines” model (pij {pj, }). • Result 1: The optimal allocation is not implementable deterministically. Best possible truthful approximation > 1.14. • Even when Lj = L, Hj = H • differentiates this case from the related machines case, another consequence of the multi-dimensionality.

  13. Main Results • Result 2: a method to convert any c-approximation algorithm for the two values case to a randomized truthful in expectation mechanism that obtains a 3c-approximation. • This is not polynomial time • Result 3: (when Lj = L, Hj = H) a deterministic, truthful, and polynomial time, 2-approximation. • Twist (novelty?) in analysis: we rely on monotonicity conditions, not on explicit price constructions. • Common for single-dimensional domains (as initiated by Myerson), but not for multi-dimensional domains.

  14. Truthfulness • Define: • An “alternatives set”, A. In our case, all possible assignments of jobs to machines. • The “type” of a player, vi : A -> R. Here vi is a negative number (the minus of a sum of several “low”s and “high”s). • Let Vi denote the domain of all valid types. • An algorithm is a function f: V1 . . .  Vn -> A. • A mechanism is a tuple M = (f, P1, , Pm),where Pi : V  R is the payment function for player i. • Dfn:Truthful Mechanisms.  vi, v-i, wi : vi(f(vi, v-i)) + Pi(vi, v-i) > vi(f(wi , v-i)) + Pi(wi, v-i) • For a given algorithm, how do we check if such prices exist? • Can we come up with an equivalent definition that does not include existential qualifiers, but, rather, only conditions on f.

  15. Weak monotonicity (W-MON) • DFN (Lavi, Mu’alem, and Nisan ‘03, Bikhchandani et. al. ‘06): Suppose f(vi, v-i)=a and f(v’i, v-i)=b. Then v’i(b) - vi(b) > v’i(a) - vi(a) • If there exist prices P such that (f,P) is truthful the f must satisfy weak-monotonicity. • If there exist such prices we say that f is implementable. • THM (Saks and Yu, 2005): In convex domains, f is implementable if and only if it is weakly monotone.

  16. What does w-mon mean in our case? H All values are for machine 1. H H L L L 1 2

  17. What does w-mon mean in our case? HL What happens if machine 1 decreases two jobs to L? HL H L L L 1 2

  18. What does w-mon mean in our case? WMON: v’i(b) - vi(b) > v’i(a) - vi(a) HL = 2(H - L) HL H => The two jobs that decreased must remain on 1. L L L 1 2

  19. What does w-mon mean in our case? And what if an outside job is decreased as well? HL HL HL L L L 1 2

  20. What does w-mon mean in our case? WMON: v’i(b) - vi(b) > v’i(a) - vi(a) HL = 2(H - L) HL HL => At least two out of the three jobs that were decreased must be assigned to 1 (doesn’t matter which two, only the numbers matter). L L L 1 2

  21. What does w-mon mean in our case? And if one job is increased from L to H? HL HL HL H L L L 1 2

  22. What does w-mon mean in our case? WMON: v’i(b) - vi(b) > v’i(a) - vi(a) HL = (H - L) HL HL => If the increased job remains on 1 then two of the jobs that were decreased must be assigned to 1, or alternatively we can “move out” the increased job and keep only one of the decreased jobs. H L L L 1 2

  23. What does w-mon mean in our case? HL etc. etc. etc. … You can now see where the monotonicity term comes from. We thus get an algorithmic condition that is equivalent to the game-theoretic definition. HL HL H L L L 1 2

  24. Cycle monotonicity • W-MON may be insufficient for implementability in non-convex domains, like our discrete scheduling domain. • Rochet (1987, JME) describes “cycle monotonicity”, which generalizes W-MON, and is equivalent to implementability on every domain (with finite alternative space). • Gui, Muller, and Vohra (2004) derive prices generically for every cycle-monotone function. • Thus any cycle monotone algorithm can be “automatically” converted to a truthful mechanism.(this can also be done for W-MON algorithms on convex domains). • That’s our way of analysis in the paper. In the talk, I will concentrate on W-MON, for the sake of simplicity.

  25. Fractional allocations • For the purpose of analysis we consider the case where jobs may be assigned fractionally: • xij denotes the fraction of job j assigned to machine i. • We have ixij = 1 for every j (every job is fully assigned). • The load of machine i is li = jxij pij • Machine i’s value is still minus her load, and her utility is still Pi - li • Cycle monotonicity is still equivalent to truthfulness and we will look for for truthful and approximately optimal fractional mechanisms. • This is just an intermediate analysis step. We do not change our actual initial goal.

  26. “Rounding” a fractional solution • We will process a fractional assignment xij to a randomized assignment: Xij = Pr(j is assigned to i). • Lavi and Swamy (2005): given a fractional truthful mechanism, if E[Xij] = xij then there exist prices such that the randomized mechanism is truthful in expectation. • Kumar, Marathe, Parthasarathy, and Srinivasan (2005): given a fractional allocation, one can construct Xij such that • E[Xij]= xij • For every i, (w.p. 1), jXij pij < jxij pij + maxj: 0 < xij < 1 pij. • Thus we move from fractional deterministic mechanisms to integral randomized mechanisms, maintaining truthfulness, and almost the same approximation guarantee.

  27. Fractional W-MON • Recall that w-mon states:f(vi, v-i)=a and f(v’i, v-i)=b => v’i(b) - vi(b) > v’i(a) - vi(a)

  28. Fractional W-MON • Recall that w-mon states:f(vi, v-i)=a and f(v’i, v-i)=b => v’i(b) - vi(b) > v’i(a) - vi(a) • In our setting, f(pi, p-i) = x and f(p’i, p-i) = x’ implies: - jx’ij p’ij

  29. Fractional W-MON • Recall that w-mon states:f(vi, v-i)=a and f(v’i, v-i)=b => v’i(b) - vi(b) > v’i(a) - vi(a) • In our setting, f(pi, p-i) = x and f(p’i, p-i) = x’ implies: - jx’ij p’ij + jx’ij pij

  30. Fractional W-MON • Recall that w-mon states:f(vi, v-i)=a and f(v’i, v-i)=b => v’i(b) - vi(b) > v’i(a) - vi(a) • In our setting, f(pi, p-i) = x and f(p’i, p-i) = x’ implies: - jx’ij p’ij + jx’ij pij>- jxij p’ij + jxij pij

  31. Fractional W-MON • Recall that w-mon states:f(vi, v-i)=a and f(v’i, v-i)=b => v’i(b) - vi(b) > v’i(a) - vi(a) • In our setting, f(pi, p-i) = x and f(p’i, p-i) = x’ implies: - jx’ij p’ij + jx’ij pij>- jxij p’ij + jxij pij or, equivalently, jx’ij (pij - p’ij) > jxij (pij -p’ij)

  32. Fractional W-MON • Recall that w-mon states:f(vi, v-i)=a and f(v’i, v-i)=b => v’i(b) - vi(b) > v’i(a) - vi(a) • In our setting, f(pi, p-i) = x and f(p’i, p-i) = x’ implies: - jx’ij p’ij + jx’ij pij>- jxij p’ij + jxij pij or, equivalently, jx’ij (pij - p’ij) > jxij (pij -p’ij) We in fact show a stronger condition: j, x’ij (pij - p’ij) >xij (pij -p’ij)

  33. Obtaining fractional cycle monotonicity Lemma: suppose that a fractional algorithm, A, satisfies: • pij = Lj => xij> 1/m • pij = Hj => xij< 1/m Then A satisfies w-mon. Proof: Suppose A(pi, p-i) = x, A(p’i, p-i) = x’. We need j: x’ij (pij - p’ij) >xij (pij -p’ij) • If pij = p’ij then clearly x’ij (pij - p’ij) = xij (pij -p’ij)

  34. Obtaining fractional cycle monotonicity Lemma: suppose that a fractional algorithm, A, satisfies: • pij = Lj => xij> 1/m • pij = Hj => xij< 1/m Then A satisfies w-mon. Proof: Suppose A(pi, p-i) = x, A(p’i, p-i) = x’. We need j: x’ij (pij - p’ij) >xij (pij -p’ij) • If pij = p’ij then clearly x’ij (pij - p’ij) = xij (pij -p’ij) • If pij > p’ij then pij = Hj and p’ij = Lj => x’ij> 1/m > xij

  35. Obtaining fractional cycle monotonicity Lemma: suppose that a fractional algorithm, A, satisfies: • pij = Lj => xij> 1/m • pij = Hj => xij< 1/m Then A satisfies w-mon. Proof: Suppose A(pi, p-i) = x, A(p’i, p-i) = x’. We need j: x’ij (pij - p’ij) >xij (pij -p’ij) • If pij = p’ij then clearly x’ij (pij - p’ij) = xij (pij -p’ij) • If pij > p’ij then pij = Hj and p’ij = Lj => x’ij> 1/m > xij • Similarly if pij < p’ij then x’ij (pij - p’ij) >xij (pij -p’ij)

  36. A fractional algorithm ALG: given any integral allocation, x*, convert it to a fractional allocation x as follows. For every (i,j) s.t. x*ij=1, do: • If pij = Hj then set xij=1/m for any i,j. • If pij = Lj then set xi’j=1/m for any i’ s.t. pi’j=Lj, and set xij so that the sum of fractions will be equal to 1.

  37. A fractional algorithm ALG: given any integral allocation, x*, convert it to a fractional allocation x as follows. For every (i,j) s.t. x*ij=1, do: • If pij = Hj then set xij=1/m for any i,j. • If pij = Lj then set xi’j=1/m for any i’ s.t. pi’j=Lj, and set xij so that the sum of fractions will be equal to 1. Properties: • W-MON follows from previous lemma. • If x* is c-approx then x is 2c-approx: each machine gets additional load which is at most the total original load times 1/m, i.e. at most the original makespan. • This converts any algorithm to a truthful fractional mechanism. With the “rounding” method, we get a randomized integral mechanism.

  38. Remarks • In particular, we can take x* to be the optimal allocation. This will give us a 3-approximation randomized integral mechanism which is truthful in expectation. • This has two drawbacks: • It is not polynomial-time • Truthfulness in expectation is weaker than deterministic truthfulness (e.g. requires assuming risk-neutrality). • With all-identical lows and highs: We get a deterministic truthful mechanism, with a better approximation ratio, 2. • Constructions and observations again use cycle monotonicity, but are completely different otherwise. • Proof: uses graph flows, is longer and less straight-forward.

  39. Summary • Study multi-dimensional scheduling, a two-fold motivation: • A natural problem, related to social choice theory, and to classic CS and OR. • Explore in general multi-dimensional mechanism design, and develop new techniques/insights. • Demonstrate how to use W-MON / Cycle-Mon to obtain positive results. • Actual results are for the “two values” case: • A general method to convert any algorithm to a truthful in expectation mechanism with almost the same approx. • A deterministic 2-approx. truthful mechanism. • OPT is not implementable, best approx > 1.14.

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