max-min fair allocation of indivisible goods
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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.
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Matching algorithm
: the set of matching edges
Without loss of generality we can assume that is a forest.
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- -
+ +-
+
Matching algorithm
: the set of matching edges
Without loss of generality we can assume that is a forest.
0
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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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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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4/5 2/9
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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”.
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