稀疏随机图中的最大权独立集与匹配。使用局部Weak Convergence方法的精确结果
Maximum Weight Independent Sets and Matchings in Sparse Random Graphs. Exact Results using the Local Weak Convergence Method
摘要：
设$G(n,c/n)$和$G_r(n)$分别是$n$ -结点稀疏随机图和$r$ -正则图，${\cal I}(n,r)$和${\cal I}(n,c)$分别是$G(n,c/n)$和$G_r(n)$中最大独立集的大小.当$c\leq e$时，可以使用Karp-Sipser算法计算${\cal I}(n,c)/n$作为$n\to\infty$的渐近值。对于随机三次图$r=3$，我们只知道$.432\leq\liminf_n {\cal I}(n,3)/n \leq \limsup_n {\cal I}(n,3)\leq .4591$具有高概率（w. h. p.）作为$n\to\infty$，如Frieze和Suen以及Bollobas分别所示。在本文中，我们假设，此外，图的节点配备有非负的权重，独立产生的根据一些共同的分布，我们认为，而不是一个独立的集的最大重量。令人惊讶的是，我们发现，对于某些权重分布，即使当$c>e$和$\lim_n {\cal I}(n,r)/n$可以精确计算某些$r\geq 2$时，也可以精确计算极限$\lim_n {\cal I}(n,c)/n$。例如，当权重以参数1、$\lim_n {\cal I}(n,2e)/n\approx .5517$和$\lim_n {\cal I}(n,3)/n\approx .6077$指数分布时。我们的结果是建立使用最近开发的局部弱收敛方法进一步减少到一定的局部最优性能所表现出的模型，我们认为。
英文摘要：
Let $G(n,c/n)$ and $G_r(n)$ be an $n$-node sparse random graph and a sparse random $r$-regular graph, respectively, and let ${\cal I}(n,r)$ and ${\cal I}(n,c)$ be the sizes of the largest independent set in $G(n,c/n)$ and $G_r(n)$. The asymptotic value of ${\cal I}(n,c)/n$ as $n\to\infty$, can be computed using the Karp-Sipser algorithm when $c\leq e$. For random cubic graphs, $r=3$, it is only known that $.432\leq\liminf_n {\cal I}(n,3)/n \leq \limsup_n {\cal I}(n,3)\leq .4591$ with high probability (w.h.p.) as $n\to\infty$, as shown by Frieze and Suen and by Bollobas, respectively. In this paper we assume in addition that the nodes of the graph are equipped with non-negative weights, independently generated according to some common distribution, and we consider instead the maximum weight of an independent set. Surprisingly, we discover that for certain weight distributions, the limit $\lim_n {\cal I}(n,c)/n$ can be computed exactly even when $c>e$, and $\lim_n {\cal I}(n,r)/n$ can be computed exactly for some $r\geq 2$. For example, when the weights are exponentially distributed with parameter 1, $\lim_n {\cal I}(n,2e)/n\approx .5517$, and $\lim_n {\cal I}(n,3)/n\approx .6077$. Our results are established using the recently developed local weak convergence method further reduced to a certain local optimality property exhibited by the models we consider.
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