平均度为d= 2 m/n的稀疏随机图G（n，m）的独立数很可能是α（G（n，m））~ 2nln（d）/d.此外，一个平凡的贪心算法w. h. p.找到一个大小为（1+o（1））n ln（d）/d的独立集，即最大大小的一半。然而，尽管有30年的广泛研究，没有有效的算法已经出现，以产生一个独立集与（1+c）n ln（d）/d，任何固定的c>0。本文证明了随机图中独立集问题的组合结构在独立集的大小k通过点knln（d）/d时发生相变。粗略地说，我们证明了大小为k>（1+c）n ln（d）/d的独立集形成了一个错综复杂的参差不齐的景观，其中局部搜索算法必然会陷入困境。我们说明了这一现象，通过提供一个指数下界的大都会过程，抽样独立集的马尔可夫链。
英文摘要：
The independence number of a sparse random graph G(n,m) of average degree d=2m/n is well-known to be α(G(n,m))~2n ln(d)/d with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1+o(1)) n ln(d)/d, i.e. half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with (1+c)n ln(d)/d, for any fixed c>0. In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point k nln(d)/d. Roughly speaking, we prove that independent sets of size k>(1+c)n ln(d)/d form an intricately ragged landscape, in which local search algorithms are bound to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independents sets.
A Note on a Nearly Uniform Partition into Common Independent Sets of Two Matroids
本说明是对K. Takazawa和Y. Yokoi（A generalized-polymatoid approach to disjoint common independent sets in two matroids，Discrete Mathematics（2019））.对于给定的$E$上的两个拟阵，在与他们论文中相同的假设下，保证存在$E$到两个拟阵的$k$公共独立集的划分，我们证明了存在$E$到$k$公共独立集的几乎一致的划分$\mathcal{P}$，其中$\mathcal{P}$中任何两个集合的基数之差最多为1。
