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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 = jSipij.
- Our goal is then to minimize the maximal load (a.k.a the “makespan” of the schedule).

Example

Two machines, three jobs:

Job p1j p2j

J1 2 1

J2 2 3

J3 3 4

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:

Job p1j p2j

J1 2 1

J2 2 3

J3 3 4

A possible assignment:

J2

J3

J1

Max. Makespan = 4

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

1

2

Example

Two machines, three jobs:

Job p1j p2j

J1 2 1

J2 2 3

J3 3 4

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 allocation monotonicity conditions, not on explicit price constructions.
- Common for single-dimensional domains (as initiated by Myerson), but not for multi-dimensional domains.

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.

Truthfulness

- Define:
- An “alternatives set”, A. In our case, all possible assignments of jobs to machines.
- The “type” of a player, ci : A -> R. Here ci is the machine’s load for the given assignment.
- 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. ci, c-i, c’i :

Pi(ci, c-i) - ci(f(ci, c-i)) > Pi(c’i, c-i) - ci(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.

Weak monotonicity (W-MON)

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

ci(b) - c’i(b) > ci(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.

What does w-mon mean in our case?

HL

What happens if machine 1 decreases two jobs to L?

HL

H

L

L

L

allocation = a

c(a) = 2L + 2H

c’(a) = 4L

1

2

What does w-mon mean in our case?

WMON: ci(b) - c’i(b) > ci(a) - c’i(a)

HL

= 2(H - L)

HL

H

- 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.

L

L

L

1

2

What does w-mon mean in our case?

WMON: ci(b) - c’i(b) > ci(a) - c’i(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?

WMON: ci(b) - c’i(b) > ci(a) - c’i(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 get an algorithmic condition instead of 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.

Fractional conditions for truthfulness

- The following condition implies w-mon (easy proof omitted):

j, x’ij (pij - p’ij) >xij (pij -p’ij)

(where f(pi, p-i) = x and f(p’i, p-i) = x’)

- In words:
- if pij > p’ij then x’ij>xij
- if pij < p’ij then x’ij<xij
- if pij = p’ij then no relation between x’ij andxij is required.

Corollary: suppose that a fractional algorithm, A, satisfies:

- pij = Lj => xij> 1/m
- pij = Hj => xij< 1/m

Then A satisfies w-mon.

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 transfer a fraction 1/m of j from i to every machine i’ with pi’j=Lj.

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 transfer a fraction 1/m of j from i to every machine i’ with pi’j=Lj.

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.

“Rounding” a fractional solution

- We will process a fractional assignment xij to a randomized assignment: Xij = random variable, indicates if i gets j.
- 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.
- 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.

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.

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 > 1.14.
- Open (and seems hard): the case of k>3 fixed values.

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