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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

We study the following KS₂(c) problem: let c ∈ ℝ^+ be some constant, and v₁,…, v_m ∈ ℝ^d be vectors such that ‖v_i‖² ≤ α for any i ∈ [m] and ∑_{i=1}^m ⟨v_i, x⟩² = 1 for any x ∈ ℝ^d with ‖x‖ = 1. The KS₂(c) problem asks to find some S ⊂ [m], such that it holds for all x ∈ ℝ^d with ‖x‖ = 1 that |∑_{i∈S} ⟨v_i, x⟩² - 1/2| ≤ c⋅√α, or report no if such S doesn't exist. Based on the work of Marcus et al. [Adam Marcus et al., 2013] and Weaver [Nicholas Weaver, 2004], the KS₂(c) problem can be seen as the algorithmic Kadison-Singer problem with parameter c ∈ ℝ^+.
Our first result is a randomised algorithm with one-sided error for the KS₂(c) problem such that (1) our algorithm finds a valid set S ⊂ [m] with probability at least 1-2/d, if such S exists, or (2) reports no with probability 1, if no valid sets exist. The algorithm has running time O(binom(m,n)⋅poly(m, d)) for n = O(d/ε² log(d) log(1/(c√α))), where ε is a parameter which controls the error of the algorithm. This presents the first algorithm for the Kadison-Singer problem whose running time is quasi-polynomial in m in a certain regime, although having exponential dependency on d. Moreover, it shows that the algorithmic Kadison-Singer problem is easier to solve in low dimensions. Our second result is on the computational complexity of the KS₂(c) problem. We show that the KS₂(1/(4√2)) problem is FNP-hard for general values of d, and solving the KS₂(1/(4√2)) problem is as hard as solving the NAE-3SAT problem.

Ben Jourdan, Peter Macgregor, and He Sun. Is the Algorithmic Kadison-Singer Problem Hard?. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jourdan_et_al:LIPIcs.ISAAC.2023.43, author = {Jourdan, Ben and Macgregor, Peter and Sun, He}, title = {{Is the Algorithmic Kadison-Singer Problem Hard?}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {43:1--43:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.43}, URN = {urn:nbn:de:0030-drops-193457}, doi = {10.4230/LIPIcs.ISAAC.2023.43}, annote = {Keywords: Kadison-Singer problem, spectral sparsification} }

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