Max-Min Fair Allocation of Indivisible Goods

Amin SaberiStanford University

Joint work with Arash Asadpour

Fair allocation

Cake cutting: Polish Mathematicians in 40’s measure theoretic

Beyond the cake: Bandwidth, links in a

network Goods in a market

Combinatorial Structure

Max-min Fair Allocation k persons and m indivisible goods : : : utility of person i for subset C of goods.

Goal: Partition the goods

Aka The Santa Claus Problem

Job scheduling: Minimizing Makespan. [Shmoys, Tardos, Lenstra 90] [Bezakova and Dani 05].

Special case:

… [Bansal and Sviridenko 06]

Integrality gap = . [Feige 06]

Our result: the first approximation algorithm for the general case. Approximation ratio

Known Results

Configuration LP

Valid Configuration for i : A bundle C of goods s.t. ui,C ¸ T.

Integrality gap:

RECALL: Our approximation ratio:

Big goods vs. small goods

For person i and good j:

Big:

Small: otherwise

Simplifying valid configurations:

One big good or A set C of at least small goods

G : the assignment graph

Matching edge: between a person and one of her big goods

Flow edge: between a person and one of her small goods

…

…

= Fraction of good j assigned to person i

matching edges and flow edges each define well -behaved polytope. But on the same vertex set!!

Outline of the Algorithm

1. Solve the Configuration LP.

2. Round the Matching edges– Randomized rounding method– Analysis

3. Rounding the flow edges to allocate the remaining goods

Outline of the Algorithm

1. Solve the Configuration LP.

2. Round the Matching edges– Randomized rounding method– Analysis

3. Rounding the flow edges to allocate the remaining goods

Matching algorithm

: the set of matching edges

Without loss of generality we can assume that is a forest.

0.2

0.7

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Matching algorithm

: the set of matching edges

Without loss of generality we can assume that is a forest.

0.2

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- -

+ +-

+

Matching algorithm

: the set of matching edges

Without loss of generality we can assume that is a forest.

0

0.9

0.1

0.6

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Matching Algorithm

Rounding method: fractional matching distribution over integral matchings

Properties1. Each vertex is saturated with probability .

2. Value of the flow bundles does not change a lot. Concentration around the mean

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0.3

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0.5

Expected value of the number of released items:(1–0.2) + (1–0.3) + (1-0.5) + (1–0.5) =

2.5

Matching Algorithm

The first objective is easy to achievee.g. von Neumann-Birkhoff decomposition

Fails to achieve the 2nd: Imposes lots of unnecessary structures. “Not random enough.”

Our approach: with respect to the constraint:

Find the distribution that maximizes the entropy

A convex program. The dual implies Optimum belongs to an exponential family:

for some

We give a simple method for finding this distribution

Matching Algorithm

Matching Algorithmvertex realization

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0.20.3

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0.2 0.1

w.p. 0.3

w.p. 0.5 w.p. 0.1

w.p. 0.1

Matching AlgorithmBayesian update

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0 0

4/5 2/9

3/9

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1

1

Matching Algorithm Bayesian update

Analysis: proof of concentration uses two important properties order independence martingale property

same holds for

Using a generalization of Azuma-Hoeffding inequality Xi is concentrated around its means

Matching Algorithm

Outline of the Algorithm

1. Solve the Configuration LP.

2. Round the Matching edges– Randomized rounding method– Analysis

3. Rounding the flow edges to allocate the remaining goods

Allocating the small goods

1- Initial allocation: Person i selects one bundle C with probability proportional to and claims all the items in the bundle

Analysis1. double counting: the expected value of the items in the

bundle is at least 2. Concentration: w.h.p. this value is

Allocating the small goods

1- Initial allocation: Person i selects one bundle C with probability proportional to and claims all the items in the bundle

2- Eliminating conflicts: every good will be allocated to one of the people who claimed it uniformly at random

Analysis– Expected number of the people who would claim it .– Using concentration: no good will be claimed more than

times.

Allocating the small goods

1- Initial allocation: person i selects one bundle C with probability proportional to and claims all the items in the bundle

2- Eliminating conflicts: every good will be allocated to one of the people who claimed it uniformly at random

Main Theorem Everybody receives a bundle with utility

Open Directions

Closing the gap between -inapproximability result and our approximation result.

Finding a -approximation schema for the case in which .

“Minimizing Envy-ratio” and “Approximate Market Equilibria”.