The Kadison-Singer Conjecture, as proved by Marcus, Spielman, and Srivastava (MSS) [Ann. Math. 182, 327-350 (2015)], has been informally thought of as a strengthening of Batson, Spielman, and Srivastava's theorem that every undirected graph has a linear-sized spectral sparsifier [SICOMP 41, 1704-1721 (2012)]. We formalize this intuition by using a corollary of the MSS result to derive the existence of spectral sparsifiers with a number of edges linear in its number of vertices for all undirected, weighted graphs. The proof consists of two steps. First, following a suggestion of Srivastava [Asia Pac. Math. Newsl. 3, 15-20 (2013)], we show the result in the special case of graphs with bounded leverage scores by repeatedly applying the MSS corollary to partition the graph, while maintaining an appropriate bound on the leverage scores of each subgraph. Then, we extend to the general case by constructing a recursive algorithm that repeatedly (i) divides edges with high leverage scores into multiple parallel edges and (ii) uses the bounded leverage score case to sparsify the resulting graph.

翻译：Marcus、Spielman和Srivastava (MSS) [Ann. Math. 182, 327-350 (2015)] 所证明的Kadison-Singer猜想，非正式地被认为是Batson、Spielman和Srivastava定理（即每个无向图都存在一个线性尺寸的谱稀疏化器）[SICOMP 41, 1704-1721 (2012)] 的加强版本。我们通过利用MSS结果的一个推论，证明了所有无向加权图均存在边数与其顶点数呈线性关系的谱稀疏化器，从而将这一直觉形式化。证明包含两个步骤。首先，遵循Srivastava的建议 [Asia Pac. Math. Newsl. 3, 15-20 (2013)]，我们在杠杆分数有界的特殊图情形下展示该结论：通过反复应用MSS推论对图进行划分，同时保持每个子图的杠杆分数处于适当界值。然后，我们将此推广至一般情形：构造一个递归算法，该算法反复执行(i) 将高杠杆分数边分解为多条平行边，以及(ii) 利用有界杠杆分数情形对所得图进行稀疏化。