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The Effect of Adding Randomly Weighted Edges
SIAM Journal on Discrete Mathematics ( IF 0.9 ) Pub Date : 2021-06-08 , DOI: 10.1137/20m1335418
Alan M. Frieze

SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1182-1200, January 2021.
We consider the following question. We have a dense regular graph $G$ with degree $\alpha n$, where $\alpha>0$ is a constant. We add $m=o(n^2)$ random edges. The edges of the augmented graph $G(m)$ are given independent edge weights $X(e),e\in E(G(m))$. We estimate the minimum weight of some specified combinatorial structures. We show that in certain cases, we can obtain the same estimate as is known for the complete graph but scaled by a factor $\alpha^{-1}$. We consider spanning trees, shortest paths, and perfect matchings in (pseudorandom) bipartite graphs.


中文翻译:

添加随机加权边的效果

SIAM Journal on Discrete Mathematics,第 35 卷,第 2 期,第 1182-1200 页,2021 年 1 月。
我们考虑以下问题。我们有一个密集的正则图 $G$,度数为 $\alpha n$,其中 $\alpha>0$ 是一个常数。我们添加 $m=o(n^2)$ 随机边。增强图 $G(m)$ 的边被赋予独立的边权重 $X(e),e\in E(G(m))$。我们估计一些指定组合结构的最小权重。我们表明,在某些情况下,我们可以获得与已知的完整图相同的估计,但按因子 $\alpha^{-1}$ 缩放。我们考虑(伪随机)二部图中的生成树、最短路径和完美匹配。
更新日期:2021-06-08
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