Max-Min Fair Allocation of Indivisible Goods Amin Saberi Stanford University Joint work with Arash Asadpour TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A

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. u i,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

Matching algorithm : the set of matching edges Without loss of generality we can assume that is a forest

Matching algorithm : the set of matching edges Without loss of generality we can assume that is a forest

Matching Algorithm Rounding method: fractional matching distribution over integral matchings Properties 1.Each vertex is saturated with probability. 2.Value of the flow bundles does not change a lot. Concentration around the mean 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 achieve e.g. von Neumann-Birkhoff decomposition Fails to achieve the 2 nd :  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 Algorithm vertex realization w.p. 0.3 w.p. 0.5w.p. 0.1

Matching Algorithm Bayesian update /5 2/9 3/9 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 X i 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 Analysis 1.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”.