An Improved Approximation Algorithm For MMS Allocation Input Output - - PowerPoint PPT Presentation

An Improved Approximation Algorithm For MMS Allocation

Input

š Agents: 𝑂 = {1,2, … , 𝑜} š Indivisible items: 𝑁 = {1, 2, … , 𝑛} š Additive valuation functions 𝑤! 𝑇 = ∑"∈$ 𝑤!" for all 𝑗 ∈ 𝑂, 𝑇 ⊆ 𝑁

Output

š A 3/4-Maximin Share (MMS) allocation 𝐵%, 𝐵&, … , 𝐵' where 𝑤!(𝐵!) ≥ ⁄

( ) 𝑁𝑁𝑇! (aka maximin value)

1

MMS value / partition / allocation

Agents\items

🍏 🍍 🍑 🍎 🥦 👪 3 1 2 5 4 👩 4 4 5 3 2 👩 value 9 9 MMS value 9

🍍 🍎🥦

👪 value 7 8 MMS value 7

🍏🍑 🍏 🥦 🍍 🍑🍎 🍍 🍎🥦

MMS allocation: 𝑤!(𝐵!) ≥ 𝑁𝑁𝑇!

👪 👩

🍎🥦 🍏 🍍🍑

2

Finding MMS value is hard!

MMS allocation might not exist, but 3/4-MMS allocation always exist

Algorithm Big Picture

To show the existence of 3/4-MMS allocation:

We assume MMSi is known for all 𝑗 ⟹ Scale valuations such that MMSi = 1 for all 𝑗 ⇒ 𝑤! 𝑁 ≥ 𝑜 š Step 1: Valid Reductions šExist 𝑇 ⊆ 𝑁 and 𝑗∗ ∈ 𝑂 such that 𝑤!∗(𝑇) ≥ ( ⁄

# $)𝑁𝑁𝑇!∗ %(𝑁)

š𝑁𝑁𝑇!

%&'(𝑁\S) ≥ 𝑁𝑁𝑇! %(𝑁) for all 𝑗 ≠ 𝑗∗

š Step 2: Generalized Bag Filling

3

Results

Existence of 3/4- MMS allocation

v

i

( M ) / n

Strongly Polynomial- time Algorithm for 3/4-MMS allocation

More careful analysis

Existence of (3/4+1/(12n))-MMS allocation

PTAS

PTAS for (3/4+1/(12n))- MMS allocation

4

MMS values are known MMS values are known