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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

1 2 1

2 2 3

3 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

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

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

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?

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?

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?

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.

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