Truthful Mechanism Design for Multi-Dimensional Scheduling via Cycle Monotonicity - PowerPoint PPT Presentation

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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. 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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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 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).

Example

Two machines, three jobs:

Jobp1jp2j

121

223

334

A possible assignment:

J2

J3

J1

1

2

Example

Two machines, three jobs:

Jobp1jp2j

121

223

334

A possible assignment:

l1 = 3

l2 = 4

J2

J3

J1

Makespan = 4

1

2

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).

Example

Two machines, three jobs:

Jobp1jp2j

121

223

334

A possible assignment:

J2

J3

J1

Max. Makespan = 4

Tot. Welfare = -3 - 3 -1 = -7

1

2

Example

Two machines, three jobs:

Jobp1jp2j

121

223

334

A possible assignment:

J2

J3

J1

Max. Makespan = 5

Tot. Welfare = -2 - 3 -1 = -6

1

2

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?

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.

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!

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.

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.

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.

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.

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.

What does w-mon mean in our case?

H

All values are for machine 1.

H

H

L

L

L

1

2

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

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

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

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

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

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

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

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.

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.

“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.

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)

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

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

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

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)

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)

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)

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

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)

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.

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.

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.

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.