最大最小公平分配(max-min fair allocation)問題是個特殊且有趣的分配
問題，此問題又稱作聖誕老人問題(the Santa Claus problem)。基本上，我
們須將 n 個不可分割資源分配給 m 位玩家，每一位玩家對於不同資源都
有各自的喜好，因此每一個資源都有 m 個非負的可能價值，我們的目標為
找到一個公平分配且極大化獲得最少資源價值的玩家。
在本研究中，我們考慮一般性的最大最小公平分配問題(the general
max-min assignment problem)，此問題可分成兩種子問題來討論，(non-
zero possible value)以及零可能價值(zero possible value)，我們同時引入一個過度估計(over-estimate)技巧，來克服處理此問題會遇到的一些障礙。在非
零可能價值的問題中，我們可以將其轉換為機台覆蓋問題(machine covering
problem)，並在多項式時間內找出$(\frac{c}{1-\epsilon})$倍的近似分配，$\epsilon$ > 0；在零可能價值的問題中，我們使用與[11, 10]相似的演算法並找出$(1+3\hat{c}+O(\delta \hat{c}^2))$倍的近似分配， $\delta$> 0，其時間複雜度為$poly(m,n)\cdot m^{poly(1/\delta)}$。其中$c$和$\hat{c}$皆來自問題給定的資源價值的最大差距，當$c$和 $\hat{c}$ 為較小的常數時，我們的演算法可以有不錯的表現。

Max-min fair allocation is an interesting assignment problem,
which maximizes the minimum total worth of resources obtained by any player.
It is also well-known as \emph{the Santa Claus problem}. Basically,
there are $m$ players and $n$ indivisible resources.
Each player has his/her own preference for the resources.
Thus, each resource has $m$ non-negative possible values.
The goal is to find an assignment such that the minimum value of the total worth of resources assigned to each player is maximized.

In this paper, we consider the general max-min assignment problem and divide it into two sub-problems, instances with non-zero values and instances in which zero values are allowed.
We also introduce a new strategy, \emph{over-estimate},
which can help overcome the challenges caused by multiple
possible values of each resource.
In the former case, we transform it into the machine covering problem and find the $(\frac{c}{1-\epsilon})$ approximation allocation in polynomial running time, where $\epsilon > 0$.
In the latter case, we present an approximation algorithm under a similar framework of Cheng et al.'s result~\cite{SWC1,SWC2}.
Our proposed algorithm can achieve
a $(1+3\hat{c}+O(\delta \hat{c}^2))$-approximation allocation
in $poly(m,n)\cdot m^{poly(1/\delta)}$ time.
Note that both $c$ and $\hat{c}$ are the constants from the given input, such as maximum gap between the largest and the smallest value over all resources.
Our results have the better performances if $c$ and $\hat{c}$ are small constants.

摘要 i
Abstract ii
誌謝 iii
List of Tables vi
List of Figures vii
1 Introduction 1
1.1 Overview of new results and techniques 5
2 The non-zero possible value instance 10
3 The zero possible value instance 12
3.1 The configuration LP 12
3.2 Binary search 13
3.3 Resources and Edges 13
3.4 Local search and bipartite graph 14
3.5 Greedy player and limited blocking 16
3.6 Over-estimate in zero value instance 16
4 An approximation algorithm for the zero value instance 17
4.1 The algorithm 17
4.2 Build phase 17
4.3 Collapse phase 20
5 Analysis 22
5.1 Invariants 22
5.2 Bounding the number of blocking edges 23
5.3 Geometric growth 24
6 Conclusion 31
References 32

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