Is the Algorithmic Kadison-Singer Problem Hard?

Authors Ben Jourdan, Peter Macgregor, He Sun

Thumbnail PDF


  • Filesize: 0.88 MB
  • 18 pages

Document Identifiers

Author Details

Ben Jourdan
  • University of Edinburgh, UK
Peter Macgregor
  • University of Edinburgh, UK
He Sun
  • University of Edinburgh, UK

Cite AsGet BibTex

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)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
  • Kadison-Singer problem
  • spectral sparsification


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Nima Anari and Shayan Oveis Gharan. The Kadison-Singer problem for strongly Rayleigh measures and applications to asymmetric TSP. CoRR, abs/1412.1143, 2014. URL:
  2. Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Nikhil Srivastava. Approximating the largest root and applications to interlacing families. In 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'18), pages 1015-1028, 2018. Google Scholar
  3. Nikhil Bansal. Constructive algorithms for discrepancy minimization. In 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS'10), pages 3-10, 2010. Google Scholar
  4. Nikhil Bansal, Tim Oosterwijk, Tjark Vredeveld, and Ruben van der Zwaan. Approximating vector scheduling: Almost matching upper and lower bounds. Algorithmica, 76(4):1077-1096, 2016. Google Scholar
  5. Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-Ramanujan sparsifiers. SIAM Journal on Computing, 41(6):1704-1721, 2012. Google Scholar
  6. Luca Becchetti, Andrea E. F. Clementi, Emanuele Natale, Francesco Pasquale, and Luca Trevisan. Finding a bounded-degree expander inside a dense one. In 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'20), pages 1320-1336, 2020. Google Scholar
  7. Peter G. Casazza. Consequences of the Marcus/Spielman/Stivastava solution to the Kadison-Singer problem. CoRR, abs/1407.4768, 2014. URL:
  8. Peter G Casazza, Matthew Fickus, Janet C Tremain, and Eric Weber. The Kadison-Singer problem in mathematics and engineering: a detailed account. Contemporary Mathematics, 414:299, 2006. Google Scholar
  9. Moses Charikar, Alantha Newman, and Aleksandar Nikolov. Tight hardness results for minimizing discrepancy. In 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'11), pages 1607-1614, 2011. Google Scholar
  10. Chandra Chekuri and Sanjeev Khanna. On multidimensional packing problems. SIAM Journal on Computing, 33(4):837-851, 2004. Google Scholar
  11. Michael B. Cohen, Cameron Musco, and Jakub Pachocki. Online row sampling. Theory of Computing, 16(15):1-25, 2020. Google Scholar
  12. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  13. Richard Kadison and Isadore Singer. Extensions of pure states. American Journal of Mathematics, 81:383-400, 1959. Google Scholar
  14. Adam Marcus, Daniel A. Spielman, and Nikhil Srivastava. Interlacing families I: bipartite Ramanujan graphs of all degrees. In 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS'13), pages 529-537, 2013. Google Scholar
  15. Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava. Interlacing families II: mixed characteristic polynomials and the Kadison-Singer problem. Annals of Mathematics, 182(1):327-350, 2015. Google Scholar
  16. Elaine Rich. Automata, computability and complexity: theory and applications. Pearson Prentice Hall Upper Saddle River, 2008. Google Scholar
  17. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM Journal on Computing, 40(6):1913-1926, 2011. Google Scholar
  18. Daniel A. Spielman and Peng Zhang. Hardness results for Weaver’s discrepancy problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM '22), pages 40:1-40:14, 2022. Google Scholar
  19. Joel A Tropp. User-friendly tail bounds for sums of random matrices. Foundations of computational mathematics, 12(4):389-434, 2012. Google Scholar
  20. Nicholas Weaver. The Kadison-Singer problem in discrepancy theory. Discrete Mathematics, 278(1-3):227-239, 2004. Google Scholar
  21. Nik Weaver. The Kadison-Singer problem in discrepancy theory, II. CoRR, abs/1303.2405, 2013. URL: