加权随机交图中的MAX CUT与稀疏随机集系统的离散化
MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems
摘要：
令$V$是$n$顶点的集合，${\cal M}$是$m$标签的集合，并且令$\mathbf{R}$是具有成功概率$p$的独立伯努利随机变量的$m \times n$矩阵。加权随机交叉图模型的随机实例$G(V,E,\mathbf{R}^T\mathbf{R})$通过在权重大于0的任意两个顶点$u,v$之间绘制具有权重$[\mathbf{R}^T\mathbf{R}]_{v,u}$的边来构造。在本文中，我们研究了加权最大割的平均情况分析，假设输入是一个加权的随机交叉图，即给定$G(V,E,\mathbf{R}^T\mathbf{R})$，我们希望找到一个分区的$V$分为两个集合，使总的重量有一个端点在每个集合是最大化的边缘。我们首先证明浓度的重量的最大削减$G(V,E,\mathbf{R}^T\mathbf{R})$周围的期望值，然后表明，当标签的数量远小于顶点的数量，一个随机分区的顶点达到渐近最优的削减重量与高概率（whp）。此外，在情况下$n=m$和恒定的平均度，我们表明，whp，多数类型的算法输出一个削减的重量是大于一个乘法常数严格大于1的随机削减的重量。然后，我们强调之间的连接的计算问题，找到一个加权最大割$G(V,E,\mathbf{R}^T\mathbf{R})$和问题，找到一个2-着色与最小的差异为一组系统$Σ$与关联矩阵$\mathbf{R}$。我们利用这种连接提出了一个（弱）bipartization算法的情况下，$m=n, p=\frac{Θ(1)}{n}$，当它终止时，它的输出可以用来找到一个2-着色$Σ$中的最小差异。最后证明了这种2-着色对应于$G(V,E,\mathbf{R}^T\mathbf{R})$中具有最大割权的双划分。
英文摘要：
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 $Σ$ with incidence matrix $\mathbf{R}$. We exploit this connection by proposing a (weak) bipartization algorithm for the case $m=n, p=\frac{Θ(1)}{n}$ that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in $Σ$. Finally, we prove that, whp this 2-coloring corresponds to a bipartition with maximum cut-weight in $G(V,E,\mathbf{R}^T\mathbf{R})$.
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