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 their 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：由Marcus、Spielman和Srivastava（MSS）[《数学年鉴》182, 327-350 (2015)]证明的Kadison-Singer猜想，非正式上被认为是Batson、Spielman和Srivastava定理（即任意无向图均存在线性规模的谱稀疏化算子[SICOMP 41, 1704-1721 (2012)]）的强化版本。我们通过利用MSS结论的一个推论来形式化这一直觉，证明了所有带权无向图均存在边数与顶点数成线性关系的谱稀疏化算子。证明分为两步。首先，遵循Srivastava[《亚太数学新闻》3, 15-20 (2013)]的建议，我们通过在保持每个子图杠杆分数适当界的前提下反复应用MSS推论对图进行划分，证明了有界杠杆分数图这一特例情形。随后，我们通过构造递归算法将其推广至一般情况：该算法反复执行(i)将高杠杆分数的边拆分为多条平行边，以及(ii)利用有界杠杆分数情形对所得图进行稀疏化。