Let $V$ be a set of $n$ vertices, ${\cal M}$ a set of $m$ labels, and let $\mathbf{R}$ be an $m \times n$ matrix of independent Bernoulli random variables with success probability $p$. A random instance $G(V,E,\mathbf{R}^T\mathbf{R})$ of the weighted random intersection graph model is constructed by drawing an edge with weight $[\mathbf{R}^T\mathbf{R}]_{v,u}$ between any two vertices $u,v$ for which this weight is larger than 0. In this paper we study the average case analysis of Weighted Max Cut, assuming the input is a weighted random intersection graph, i.e. given $G(V,E,\mathbf{R}^T\mathbf{R})$ we wish to find a partition of $V$ into two sets so that the total weight of the edges having one endpoint in each set is maximized. We initially prove concentration of the weight of a maximum cut of $G(V,E,\mathbf{R}^T\mathbf{R})$ around its expected value, and then show that, when the number of labels is much smaller than the number of vertices, a random partition of the vertices achieves asymptotically optimal cut weight with high probability (whp). Furthermore, in the case $n=m$ and constant average degree, we show that whp, a majority type algorithm outputs a cut with weight that is larger than the weight of a random cut by a multiplicative constant strictly larger than 1. Then, we highlight a connection between the computational problem of finding a weighted maximum cut in $G(V,E,\mathbf{R}^T\mathbf{R})$ and the problem of finding a 2-coloring with minimum discrepancy for a set system $\Sigma$ with incidence matrix $\mathbf{R}$. We exploit this connection by proposing a (weak) bipartization algorithm for the case $m=n, p=\frac{\Theta(1)}{n}$ that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in $\Sigma$. Finally, we prove that, whp this 2-coloring corresponds to a bipartition with maximum cut-weight in $G(V,E,\mathbf{R}^T\mathbf{R})$.

中文翻译：

---

加权随机交图中的 MAX CUT 和稀疏随机集系统的差异

令 $V$ 是一组 $n$ 个顶点，${\cal M}$ 是一组 $m$ 标签，让 $\mathbf{R}$ 是一个 $m \times n$ 独立伯努利随机矩阵成功概率为 $p$ 的变量。加权随机相交图模型的随机实例 $G(V,E,\mathbf{R}^T\mathbf{R})$ 是通过绘制一条权重为 $[\mathbf{R}^T\mathbf 的边来构建的{R}]_{v,u}$ 之间的任何两个顶点 $u,v$ 之间的权重大于 0。 本文我们研究加权最大割的平均案例分析，假设输入是加权随机交集图，即给定 $G(V,E,\mathbf{R}^T\mathbf{R})$ 我们希望找到 $V$ 的分区为两组，以便具有一个端点的边的总权重在每个集合中最大化。我们最初证明了 $G(V,E, \mathbf{R}^T\mathbf{R})$ 围绕其期望值，然后证明，当标签数量远小于顶点数量时，顶点的随机分区实现渐近最优切割权重为高概率（whp）。此外，在 $n=m$ 和恒定平均度的情况下，我们表明 whp，多数类型算法输出权重大于随机切割的权重，乘法常数严格大于 1。然后，我们强调了在 $G(V,E,\mathbf{R}^T\mathbf{R})$ 中找到加权最大割的计算问题与找到具有最小差异的 2-coloring 的问题之间的联系用关联矩阵 $\mathbf{R}$ 设置系统 $\Sigma$。我们通过为 $m=n 的情况提出（弱）二分算法来利用这种联系，p=\frac{\Theta(1)}{n}$，当它终止时，它的输出可用于在 $\Sigma$ 中找到具有最小差异的 2-coloring。最后，我们证明，whp 这个 2-coloring 对应于 $G(V,E,\mathbf{R}^T\mathbf{R})$ 中最大切权重的二分。