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تسجيل مستخدم جديدTight local approximation results for max-min linear programs
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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 $epsilon>0$ there exists a local algorithm achieving the approximation ratio ${Delta_I (1 - 1/Delta_K)} + epsilon$. We also show that this result is the best possible -- no local algorithm can achieve the approximation ratio ${Delta_I (1 - 1/Delta_K)}$. Here $Delta_I$ is the maximum degree of a vertex $i in I$, and $Delta_K$ 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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