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

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

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Example Jobp 1j p 2j J121 J223 J334 Two machines, three jobs: A possible assignment: 12 J3 J1 J2

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Example Jobp 1j p 2j J121 J223 J334 Two machines, three jobs: A possible assignment: 12 J3 J1 J2 Makespan = 4 l 1 = 3l 2 = 4

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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 p ij for executing the job. A machine may get a payment, P i, and its total utility is: P i - l i 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).

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Example Jobp 1j p 2j J121 J223 J334 Two machines, three jobs: A possible assignment: 12 J3 J1 J2 Max. Makespan = 4 Tot. Welfare = = -7

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Example Jobp 1j p 2j J121 J223 J334 Two machines, three jobs: A possible assignment: 12 J3 J1 J2 Max. Makespan = 5 Tot. Welfare = = -6

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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?

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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. Why is this question important? (2)

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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!

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Current status (2) Archer and Tardos (2001) study the special case of related machines: each machine has speed s i, and p ij = p j /s i. –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.

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

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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 L j = L, H j = H) a deterministic, truthful, and polynomial time, 2-approximation. Twist (novelty?) in analysis: we rely on allocation monotonicity conditions, not on explicit price constructions. –Common for single-dimensional domains (as initiated by Myerson), but not for multi-dimensional domains.

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The Randomized Construction: Outline Description of the monotonicity conditions that are sufficient to obtain truthfulness. A transition to a fractional domain. Achieving monotonicity and makespan approximation in the fractional domain. Rounding the fractional solution.

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Truthfulness Define: –An “alternatives set”, A. In our case, all possible assignments of jobs to machines. –The “type” of a player, c i : A -> R. Here c i is the machine’s load for the given assignment. –Let V i denote the domain of all valid types. –An algorithm is a function f: V 1 ... V n -> A. A mechanism is a tuple M = (f, P 1, , P m ), where P i : V R is the payment function for player i. Dfn: Truthful Mechanisms. c i, c -i, c’ i : P i (c i, c -i ) - c i (f(c i, c -i )) > P i (c’ i, c -i ) - c i (f(c’ i, c -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.

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Weak Monotonicity

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Weak monotonicity (W-MON) DFN (Lavi, Mu’alem, and Nisan ‘03, Bikhchandani et. al. ‘06): Suppose f(c i, c -i )=a and f(c’ i, c -i )=b. Then c i (b) - c’ i (b) > c i (a) - c’ i (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.

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12 L L H What does w-mon mean in our case? L H H All costs are for machine 1 allocation = a c(a) = 2L + 2H

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12 L L H What does w-mon mean in our case? L HLHL HLHL What happens if machine 1 decreases two jobs to L? allocation = a c(a) = 2L + 2H c’(a) = 4L

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12 L L H What does w-mon mean in our case? L HLHL HLHL WMON: c i (b) - c’ i (b) > c i (a) - c’ i (a) = 2(H - L) What allocations (b) satisfy the inequality? The only way to get at least 2(H - L) on the left side is to assign the two the two jobs that were decreased to machine 1.

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12 L L HLHL What does w-mon mean in our case? L HLHL HLHL And what if an outside job is decreased as well?

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12 L L HLHL What does w-mon mean in our case? L HLHL HLHL WMON: c i (b) - c’ i (b) > c i (a) - c’ i (a) = 2(H - L) => 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).

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12 L L HLHL What does w-mon mean in our case? L HLHL HLHL And if one job is increased from L to H? H

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12 L L HLHL What does w-mon mean in our case? L HLHL HLHL WMON: c i (b) - c’ i (b) > c i (a) - c’ i (a) = (H - L) => 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

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12 L L HLHL What does w-mon mean in our case? L HLHL HLHL H etc. etc. etc. … You can now see where the monotonicity term comes from. We get an algorithmic condition instead of the game-theoretic definition.

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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.

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A fractional solution

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Fractional allocations For the purpose of analysis we consider the case where jobs may be assigned fractionally: –x ij denotes the fraction of job j assigned to machine i. –We have i x ij = 1 for every j (every job is fully assigned). –The load of machine i is l i = j x ij p ij –Machine i’s value is still minus her load, and her utility is still P i - l i 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.

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

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Fractional W-MON Recall that w-mon states: f(v i, v -i )=a and f(v’ i, v -i )=b => v’ i (b) - v i (b) > v’ i (a) - v i (a) In our setting, f(p i, p -i ) = x and f(p’ i, p -i ) = x’ implies: - j x’ ij p’ ij

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Fractional W-MON Recall that w-mon states: f(v i, v -i )=a and f(v’ i, v -i )=b => v’ i (b) - v i (b) > v’ i (a) - v i (a) In our setting, f(p i, p -i ) = x and f(p’ i, p -i ) = x’ implies: - j x’ ij p’ ij + j x’ ij p ij

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Fractional W-MON Recall that w-mon states: f(v i, v -i )=a and f(v’ i, v -i )=b => v’ i (b) - v i (b) > v’ i (a) - v i (a) In our setting, f(p i, p -i ) = x and f(p’ i, p -i ) = x’ implies: - j x’ ij p’ ij + j x’ ij p ij > - j x ij p’ ij + j x ij p ij

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Fractional W-MON Recall that w-mon states: f(v i, v -i )=a and f(v’ i, v -i )=b => v’ i (b) - v i (b) > v’ i (a) - v i (a) In our setting, f(p i, p -i ) = x and f(p’ i, p -i ) = x’ implies: - j x’ ij p’ ij + j x’ ij p ij > - j x ij p’ ij + j x ij p ij or, equivalently, j x’ ij (p ij - p’ ij ) > j x ij (p ij -p’ ij )

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Fractional W-MON Recall that w-mon states: f(v i, v -i )=a and f(v’ i, v -i )=b => v’ i (b) - v i (b) > v’ i (a) - v i (a) In our setting, f(p i, p -i ) = x and f(p’ i, p -i ) = x’ implies: - j x’ ij p’ ij + j x’ ij p ij > - j x ij p’ ij + j x ij p ij or, equivalently, j x’ ij (p ij - p’ ij ) > j x ij (p ij -p’ ij ) We in fact show a stronger condition: j, x’ ij (p ij - p’ ij ) > x ij (p ij -p’ ij )

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Fractional conditions for truthfulness The following condition implies w-mon (easy proof omitted): j, x’ ij (p ij - p’ ij ) > x ij (p ij -p’ ij ) (where f(p i, p -i ) = x and f(p’ i, p -i ) = x’) In words: –if p ij > p’ ij then x’ ij > x ij –if p ij < p’ ij then x’ ij < x ij –if p ij = p’ ij then no relation between x’ ij and x ij is required. Corollary: suppose that a fractional algorithm, A, satisfies: p ij = L j => x ij > 1/m p ij = H j => x ij < 1/m Then A satisfies w-mon.

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Obtaining fractional cycle monotonicity Lemma: suppose that a fractional algorithm, A, satisfies: p ij = L j => x ij > 1/m p ij = H j => x ij < 1/m Then A satisfies w-mon. Proof: Suppose A(p i, p -i ) = x, A(p’ i, p -i ) = x’. We need j: x’ ij (p ij - p’ ij ) > x ij (p ij -p’ ij ) If p ij = p’ ij t hen clearly x’ ij (p ij - p’ ij ) = x ij (p ij -p’ ij )

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Obtaining fractional cycle monotonicity Lemma: suppose that a fractional algorithm, A, satisfies: p ij = L j => x ij > 1/m p ij = H j => x ij < 1/m Then A satisfies w-mon. Proof: Suppose A(p i, p -i ) = x, A(p’ i, p -i ) = x’. We need j: x’ ij (p ij - p’ ij ) > x ij (p ij -p’ ij ) If p ij = p’ ij t hen clearly x’ ij (p ij - p’ ij ) = x ij (p ij -p’ ij ) If p ij > p’ ij t hen p ij = H j and p’ ij = L j => x’ ij > 1/m > x ij

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Obtaining fractional cycle monotonicity Lemma: suppose that a fractional algorithm, A, satisfies: p ij = L j => x ij > 1/m p ij = H j => x ij < 1/m Then A satisfies w-mon. Proof: Suppose A(p i, p -i ) = x, A(p’ i, p -i ) = x’. We need j: x’ ij (p ij - p’ ij ) > x ij (p ij -p’ ij ) If p ij = p’ ij t hen clearly x’ ij (p ij - p’ ij ) = x ij (p ij -p’ ij ) If p ij > p’ ij t hen p ij = H j and p’ ij = L j => x’ ij > 1/m > x ij Similarly if p ij x ij (p ij -p’ ij )

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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 p ij = H j then set x ij =1/m for any i,j. If p ij = L j then transfer a fraction 1/m of j from i to every machine i’ with p i’j =L j.

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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 p ij = H j then set x ij =1/m for any i,j. If p ij = L j then transfer a fraction 1/m of j from i to every machine i’ with p i’j =L j. 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.

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Back to Integral Mechanisms

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“Rounding” a fractional solution We will process a fractional assignment x ij to a randomized assignment: X ij = random variable, indicates if i gets j. Lavi and Swamy (2005): given a fractional truthful mechanism, if E[X ij ] = x ij 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 X ij such that –E[X ij ]= x ij –For every i, (w.p. 1), j X ij p ij < j x ij p ij + max j: 0 < x ij < 1 p ij. 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.

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Remarks 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 show 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.

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Summary A two-fold motivation for multi-dimensional scheduling: –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. We 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 > Open (and seems hard): the case of k>3 fixed values.

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