极大极小线性规划的紧局部逼近结果
Tight local approximation results for max-min linear programs
摘要：
在一个二分最大-最小LP中，我们给出一个二分图$\myG = (V \cup I \cup K, E)$，其中每个代理$v \in V$与恰好一个约束$i \in I$和恰好一个目标$k \in K$相邻。每个代理$v$控制一个变量$x_v$。对于每个$i \in I$，我们有一个非负的线性约束的变量相邻的代理。对于每个$k \in K$，我们有一个非负的线性目标函数的变量相邻的代理。任务是最大化目标函数的最小值。我们研究本地算法，每个代理$v$必须选择$x_v$输入的基础上，其恒定半径的邻域内的$\myG$。我们表明，对于每一个$ε>0$存在一个本地算法实现的近似比${Δ_I (1 - 1/Δ_K)} + ε$。我们还表明，这个结果是最好的可能-没有本地算法可以实现的近似比${Δ_I (1 - 1/Δ_K)}$。这里，$Δ_I$是顶点$i \in I$的最大度，$Δ_K$是顶点$k \in K$的最大度。作为一种方法的贡献，我们介绍了局部逼近算法的设计图展开技术。
英文摘要：
In a bipartite max-min LP, we are given a bipartite graph $\myG = (V \cup I \cup K, E)$, where each agent $v \in V$ is adjacent to exactly one constraint $i \in I$ and exactly one objective $k \in K$. Each agent $v$ controls a variable $x_v$. For each $i \in I$ we have a nonnegative linear constraint on the variables of adjacent agents. For each $k \in K$ we have a nonnegative linear objective function of the variables of adjacent agents. The task is to maximise the minimum of the objective functions. We study local algorithms where each agent $v$ must choose $x_v$ based on input within its constant-radius neighbourhood in $\myG$. We show that for every $ε>0$ there exists a local algorithm achieving the approximation ratio ${Δ_I (1 - 1/Δ_K)} + ε$. We also show that this result is the best possible -- no local algorithm can achieve the approximation ratio ${Δ_I (1 - 1/Δ_K)}$. Here $Δ_I$ is the maximum degree of a vertex $i \in I$, and $Δ_K$ is the maximum degree of a vertex $k \in K$. As a methodological contribution, we introduce the technique of graph unfolding for the design of local approximation algorithms.
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