An Improved Approximation Algorithm For MMS Allocation Input Output Β Agents: π = {1,2, β¦ , π} Β A 3/4-Maximin Share (MMS) allocation π΅ % , π΅ & , β¦ , π΅ ' where Β Indivisible items: π = {1, 2, β¦ , π} ( ) πππ ! (aka maximin value) π€ ! (π΅ ! ) β₯ β Β Additive valuation functions π€ ! π = β "β$ π€ !" for all π β π , π β π 1
MMS value / partition / allocation MMS allocation: π π π π π₯¦ π€ ! (π΅ ! ) β₯ πππ ! Agents\items πͺ 3 1 2 5 4 π© 4 4 5 3 2 πͺ ππ₯¦ π π π© π π ππ πͺ π© π π₯¦ ππ ππ₯¦ ππ₯¦ ππ value 7 8 value 9 9 MMS allocation might MMS value 7 MMS value 9 not exist, but 3/4-MMS 2 allocation always exist Finding MMS value is hard!
Algorithm Big Picture To show the existence of 3/4-MMS allocation: We assume MMS i is known for all π βΉ Scale valuations such that MMS i = 1 for all π β π€ ! π β₯ π Β Step 1: Valid Reductions Β Exist π β π and π β β π such that π€ ! β (π) β₯ ( β % (π) # $ )πππ ! β %&' (π\S) β₯ πππ ! % (π) for all π β π β Β πππ ! Β Step 2: Generalized Bag Filling 3
Results Strongly Polynomial- time Algorithm for 3/4-MMS allocation n / ) M ( v i Existence of 3/4- MMS allocation More careful analysis MMS values are known Existence of PTAS for (3/4+1/(12n))- (3/4+1/(12n))-MMS PTAS MMS allocation allocation 4 MMS values are known
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