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Computer Science > Data Structures and Algorithms

arXiv:2308.12483 (cs)
[Submitted on 24 Aug 2023 (v1), last revised 9 Nov 2023 (this version, v2)]

Title:Linear-Sized Spectral Sparsifiers and the Kadison-Singer Problem

Authors:Phevos Paschalidis, Ashley Zhuang
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Abstract: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.
Subjects: Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
Cite as: arXiv:2308.12483 [cs.DS]
  (or arXiv:2308.12483v2 [cs.DS] for this version)
  https://doi.org/10.48550/arXiv.2308.12483
Related DOI: https://doi.org/10.1137/1.9781611977936.2

Submission history

From: Phevos Paschalidis [view email]
[v1] Thu, 24 Aug 2023 00:49:59 UTC (19 KB)
[v2] Thu, 9 Nov 2023 19:12:13 UTC (23 KB)
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