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تسجيل مستخدم جديدLocal approximation algorithms for a class of 0/1 max-min linear programs
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We study the applicability of distributed, local algorithms to 0/1 max-min LPs where the objective is to maximise ${min_k sum_v c_{kv} x_v}$ subject to ${sum_v a_{iv} x_v le 1}$ for each $i$ and ${x_v ge 0}$ for each $v$. Here $c_{kv} in {0,1}$, $a_{iv} in {0,1}$, and the support sets ${V_i = {v : a_{iv} > 0 }}$ and ${V_k = {v : c_{kv}>0 }}$ have bounded size; in particular, we study the case $|V_k| le 2$. Each agent $v$ is responsible for choosing the value of $x_v$ based on information within its constant-size neighbourhood; the communication network is the hypergraph where the sets $V_k$ and $V_i$ constitute the hyperedges. We present a local approximation algorithm which achieves an approximation ratio arbitrarily close to the theoretical lower bound presented in prior work.
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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 i
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A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min
LPs where the objective is to maximise $min_k sum_v c_{kv} x_v$ subject to $sum_v a_{iv} x_v le 1$ for each $i$ and $x_v ge 0$ for each $v$. Here $c_{kv} ge 0$, $a_{iv} ge 0$, and the support sets $V_i = {v : a_{iv} > 0 }$, $V_k = {v : c_{kv}>0 }$, $I_v = {i : a_{iv} > 0 }$ and $K_v = {k : c_{kv} > 0 }$ have bounded size. In the distributed setting, each agent $v$ is responsible for choosing the value of $x_v$, and the communication network is a hypergraph $mathcal{H}$ where the sets $V_k$ and $V_i$ constitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if $|V_i|$ and $|V_k|$ are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in $mathcal{H}$.
We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The objective is to maximise $omega$ subject to $Ax le 1$, $Cx ge omega 1$, and $x ge 0$ for nonnegative matrices $A$ and $C$. The approximation ratio o
f our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.
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