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6974 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 69 of 349

  • The length of an s-increasing sequence of r-tuples

  • Part of
  • W. T. Gowers, J. Long
  • Journal:
    Combinatorics, Probability and Computing / Volume 30 / Issue 5 / September 2021
    Published online by Cambridge University Press:
    08 January 2021, pp. 686-721

  • We prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if ai < bi for at least two values of i, and define a sequence a1, …, am of triples to be 2-increasing if ar is 2-less than as whenever r < s. Loh asks how long a 2-increasing sequence can be if all the triples take values in {1, 2, …, n}, and gives a log* improvement over the trivial upper bound of n2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n3/2. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well-known problems in extremal combinatorics, and asking a number of further questions.

  • Cycle partitions of regular graphs

  • Part of
  • Vytautas Gruslys, Shoham Letzter
  • Journal:
    Combinatorics, Probability and Computing / Volume 30 / Issue 4 / July 2021
    Published online by Cambridge University Press:
    18 December 2020, pp. 526-549

  • Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$, improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

Beyond the Worst-Case Analysis of Algorithms
  • Edited by Tim Roughgarden
  • Published online:
    17 December 2020
    Print publication:
    14 January 2021
  • There are no silver bullets in algorithm design, and no single algorithmic idea is powerful and flexible enough to solve every computational problem. Nor are there silver bullets in algorithm analysis, as the most enlightening method for analyzing an algorithm often depends on the problem and the application. However, typical algorithms courses rely almost entirely on a single analysis framework, that of worst-case analysis, wherein an algorithm is assessed by its worst performance on any input of a given size. The purpose of this book is to popularize several alternatives to worst-case analysis and their most notable algorithmic applications, from clustering to linear programming to neural network training. Forty leading researchers have contributed introductions to different facets of this field, emphasizing the most important models and results, many of which can be taught in lectures to beginning graduate students in theoretical computer science and machine learning.

  • Counting Hamilton cycles in Dirac hypergraphs

  • Part of
  • Stefan Glock, Stephen Gould, Felix Joos, Daniela Kühn, Deryk Osthus
  • Journal:
    Combinatorics, Probability and Computing / Volume 30 / Issue 4 / July 2021
    Published online by Cambridge University Press:
    17 December 2020, pp. 631-653

  • A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$, every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$. As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$-cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$, which makes progress on a question of Ferber, Krivelevich and Sudakov.

Page 69 of 349