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6898 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 2 of 345

  • 1 - Introduction

  • Jan Krajíček, Charles University, Prague
  • Book:
    Proof Complexity Generators
    Published online:
    29 May 2025
    Print publication:
    26 June 2025, pp 1-5

  • Summary

    The chapter gives the historic background in bounded arithmetic and describes how it lead to the development of the presented theory. It lists prerequisites and some notation and terminology to be used.

  • 3 - τ-Formulas and Generators

  • Jan Krajíček, Charles University, Prague
  • Book:
    Proof Complexity Generators
    Published online:
    29 May 2025
    Print publication:
    26 June 2025, pp 16-29

  • Summary

    The chapter defines the notion of a generator and its hardness, and formulates the hardness conjecture. It also defines a stronger notion of pseudosurjectivity of a generator and states the key conjecture about it. It examines some consequences of the two conjectures for the dWPHP problem. It also relates the hardness conjecture to feasible interpolation, gives a model-theoretic view of the issues and discusses a relation to pseudorandomness.

  • 8 - Consistency Results

  • Jan Krajíček, Charles University, Prague
  • Book:
    Proof Complexity Generators
    Published online:
    29 May 2025
    Print publication:
    26 June 2025, pp 75-92

  • Summary

    The chapter gives several consistency results related to the dWPHP problem. It also considers the hardness conjecture for feasibly infinite NP sets. It relates witnessing of dWPHP to various computational complexity conjectures.

  • 10 - Further Research

  • Jan Krajíček, Charles University, Prague
  • Book:
    Proof Complexity Generators
    Published online:
    29 May 2025
    Print publication:
    26 June 2025, pp 104-110

  • Summary

    This final chapter offers a number of topics for further research involving ordinary PHP, S-T computations, a new notion of PLS-infinite NP sets, proof search algorithms, an exponential time weakening of generators and the function inversion problem.

  • 7 - The Case of ER

  • Jan Krajíček, Charles University, Prague
  • Book:
    Proof Complexity Generators
    Published online:
    29 May 2025
    Print publication:
    26 June 2025, pp 58-74

  • Summary

    The chapter concentrates on the pivotal case of extended resolution. It recalls some characterizations of its lengths-of-proofs function and formulates a framework for lower bounds proofs using expansions of pseudofinite structures. It gives an example of a specific candidate construction.

  • 9 - Contexts

  • Jan Krajíček, Charles University, Prague
  • Book:
    Proof Complexity Generators
    Published online:
    29 May 2025
    Print publication:
    26 June 2025, pp 93-103

  • Summary

    The chapter presents several topics outside of the theory where some ideas and results around proof complexity generators appear to be relevant. These include SAT algorithms, the optimality problem of proof complexity, structured PHP approach, the incompleteness theorem and total NP search problems.

  • On the size of temporal cliques in subcritical random temporal graphs

  • Part of
  • Caelan Atamanchuk, Luc Devroye, Gábor Lugosi
  • Journal:
    Combinatorics, Probability and Computing , First View
    Published online by Cambridge University Press:
    26 June 2025, pp. 1-9
    • Article
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  • A random temporal graph is an Erdős-Rényi random graph $G(n,p)$, together with a random ordering of its edges. A path in the graph is called increasing if the edges on the path appear in increasing order. A set $S$ of vertices forms a temporal clique if for all $u,v \in S$, there is an increasing path from $u$ to $v$. Becker, Casteigts, Crescenzi, Kodric, Renken, Raskin and Zamaraev [(2023) Giant components in random temporal graphs. arXiv,2205.14888] proved that if $p=c\log n/n$ for $c\gt 1$, then, with high probability, there is a temporal clique of size $n-o(n)$. On the other hand, for $c\lt 1$, with high probability, the largest temporal clique is of size $o(n)$. In this note, we improve the latter bound by showing that, for $c\lt 1$, the largest temporal clique is of constant size with high probability.

  • 4 - The Stretch

  • Jan Krajíček, Charles University, Prague
  • Book:
    Proof Complexity Generators
    Published online:
    29 May 2025
    Print publication:
    26 June 2025, pp 30-41

  • Summary

    The chapter examines possible stretch of generators and relates it to problems about Kolmogorov's complexity, the feasible disjunction property and to a related notion of disjunction hardness of generators. The truth-table function is presented as a key example of a generator and its hardness and pseudosurjectivity is considered. A problem about time-bounded Kolmogorov complexity is formulated.

Oriented Matroids
  • Laura Anderson
  • Published online:
    04 June 2025
    Print publication:
    10 April 2025
  • Oriented matroids appear throughout discrete geometry, with applications in algebra, topology, physics, and data analysis. This introduction to oriented matroids is intended for graduate students, scientists wanting to apply oriented matroids, and researchers in pure mathematics. The presentation is geometrically motivated and largely self-contained, and no knowledge of matroid theory is assumed. Beginning with geometric motivation grounded in linear algebra, the first chapters prove the major cryptomorphisms and the Topological Representation Theorem. From there the book uses basic topology to go directly from geometric intuition to rigorous discussion, avoiding the need for wider background knowledge. Topics include strong and weak maps, localizations and extensions, the Euclidean property and non-Euclidean properties, the Universality Theorem, convex polytopes, and triangulations. Themes that run throughout include the interplay between combinatorics, geometry, and topology, and the idea of oriented matroids as analogs to vector spaces over the real numbers and how this analogy plays out topologically.

Games of No Chance 4
  • Edited by Richard J. Nowakowski
  • Mathematical Sciences Research Institute
  • Published online:
    30 May 2025
    Print publication:
    16 April 2015
  • Combinatorial games are the strategy games that people like to play, for example chess, Hex, and Go. They differ from economic games in that there are two players who play alternately with no hidden cards and no dice. These games have a mathematical structure that allows players to analyse them in the abstract. Games of No Chance 4 contains the first comprehensive explorations of misère (last player to move loses) games, extends the theory for some classes of normal-play (last player to move wins) games and extends the analysis for some specific games. It includes a tutorial for the very successful approach to analysing misère impartial games and the first attempt at using it for misère partisan games. Hex and Go are featured, as well as new games: Toppling Dominoes and Maze. Updated versions of Unsolved Problems in Combinatorial Game Theory and the Combinatorial Games Bibliography complete the volume.

Algorithmic Number Theory
  • Lattices, Number Fields, Curves and Cryptography
  • Edited by J. P. Buhler, P. Stevenhagen
  • Mathematical Sciences Research Institute
  • Published online:
    30 May 2025
    Print publication:
    20 October 2008
  • Number theory is one of the oldest and most appealing areas of mathematics. Computation has always played a role in number theory, a role which has increased dramatically in the last 20 or 30 years, both because of the advent of modern computers, and because of the discovery of surprising and powerful algorithms. As a consequence, algorithmic number theory has gradually emerged as an important and distinct field with connections to computer science and cryptography as well as other areas of mathematics. This text provides a comprehensive introduction to algorithmic number theory for beginning graduate students, written by the leading experts in the field. It includes several articles that cover the essential topics in this area, and in addition, there are contributions pointing in broader directions, including cryptography, computational class field theory, zeta functions and L-series, discrete logarithm algorithms, and quantum computing.

  • Tight Hamilton cycles with high discrepancy

  • Part of
  • Lior Gishboliner, Stefan Glock, Amedeo Sgueglia
  • Journal:
    Combinatorics, Probability and Computing / Volume 34 / Issue 4 / July 2025
    Published online by Cambridge University Press:
    30 May 2025, pp. 565-584
    • Article
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      • Open access
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  • In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$ with minimum $(k-1)$-degree $\delta (G) \ge (1/2+o(1))n$ contains a tight Hamilton cycle with high discrepancy, that is, with at least $n/r+\Omega (n)$ edges of one colour. The minimum degree condition is asymptotically best possible and our theorem also implies a corresponding result for perfect matchings. Our tools combine various structural techniques such as Turán-type problems and hypergraph shadows with probabilistic techniques such as random walks and the nibble method. We also propose several intriguing problems for future research.

Page 2 of 345