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6950 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 43 of 348

  • Sparse recovery properties of discrete random matrices

  • Part of
  • Asaf Ferber, Ashwin Sah, Mehtaab Sawhney, Yizhe Zhu
  • Journal:
    Combinatorics, Probability and Computing / Volume 32 / Issue 2 / March 2023
    Published online by Cambridge University Press:
    04 October 2022, pp. 316-325
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  • Motivated by problems from compressed sensing, we determine the threshold behaviour of a random $n\times d \pm 1$ matrix $M_{n,d}$ with respect to the property ‘every $s$ columns are linearly independent’. In particular, we show that for every $0\lt \delta \lt 1$ and $s=(1-\delta )n$, if $d\leq n^{1+1/2(1-\delta )-o(1)}$ then with high probability every $s$ columns of $M_{n,d}$ are linearly independent, and if $d\geq n^{1+1/2(1-\delta )+o(1)}$ then with high probability there are some $s$ linearly dependent columns.

  • Improved bound for improper colourings of graphs with no odd clique minor

  • Part of
  • Raphael Steiner
  • Journal:
    Combinatorics, Probability and Computing / Volume 32 / Issue 2 / March 2023
    Published online by Cambridge University Press:
    30 September 2022, pp. 326-333
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  • Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$-minor admits a vertex $(2t-2)$-colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$. The bound on the number of colours is optimal up to a factor of $2$, improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput. 17 815–821), Kang and Oum (2019, Combin. Probab. Comput. 28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc. 98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$-minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$-minor-free graphs with some additional ideas, which make the extension to odd $K_t$-minor-free graphs possible.

  • 8 - First-order formulas for fundamental sequence properties

  • Jeffrey Shallit, University of Waterloo, Ontario
  • Book:
    The Logical Approach to Automatic Sequences
    Published online:
    09 September 2022
    Print publication:
    29 September 2022, pp 113-192

  • Summary

    In this chapter we will examine about 80 different fundamental properties of automatic sequences, and show how each one can be encoded by first-order logical formulas. We then use Walnut to re-derive new proofs of known results, or prove new results, concerning some famous automatic sequences. You can use these examples to learn what Walnut is capable of, but also as a ‘catalogue’ of first-order statements of fundamental properties of sequences.

  • 6 - First-order logic and automatic sequences

  • Jeffrey Shallit, University of Waterloo, Ontario
  • Book:
    The Logical Approach to Automatic Sequences
    Published online:
    09 September 2022
    Print publication:
    29 September 2022, pp 82-95

  • Summary

    In this chapter we give yet another fundamental way to think about automatic sequences, based on first-order logic. This revolutionary approach is originally due to Büuchi, with elaborations and additions by Bruyére, Hansel, Michaux, and Villemaire, and their ideas form the basis for this book. A good reference for the material in this section is the wonderful survey paper by these last four authors [56].

  • 7 - Using Walnut

  • Jeffrey Shallit, University of Waterloo, Ontario
  • Book:
    The Logical Approach to Automatic Sequences
    Published online:
    09 September 2022
    Print publication:
    29 September 2022, pp 96-112

  • Summary

    Walnut is free software originally designed and written in Java by Hamoon Mousavi [278], and recently modified by Aseem Raj Baranwal, Laindon C. Burnett, Kai Hsiang Yang, and Anatoly Zavyalov. This book is based on the most recent version of Walnut, called Walnut 3.7, which is available for free download at

  • 5 - Automatic sequences

  • Jeffrey Shallit, University of Waterloo, Ontario
  • Book:
    The Logical Approach to Automatic Sequences
    Published online:
    09 September 2022
    Print publication:
    29 September 2022, pp 62-81

  • Summary

    In this chapter we introduce the main subject of the book, which is automatic sequences. Roughly speaking, an automatic sequence is a sequence over a finite alphabet whose nth term can be computed by a finite automaton reading a representation for n in a regular numeration system (Section 6.4).

Page 43 of 348