922 research outputs found

    Algorithms and complexity for approximately counting hypergraph colourings and related problems

    Get PDF
    The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows. • When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard. • When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

    Full text link
    Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lov\'asz Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index

    Product structure of graph classes with strongly sublinear separators

    Full text link
    We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class G\mathcal{G} admits O(n1ϵ)O(n^{1-\epsilon}) separators, then for any fixed δ(0,ϵ)\delta\in(0,\epsilon) every nn-vertex graph in G\mathcal{G} is a subgraph of the strong product of a graph HH with bounded tree-depth and a complete graph of size O(n1ϵ+δ)O(n^{1-\epsilon+\delta}). This result holds with δ=0\delta=0 if we allow HH to have tree-depth O(loglogn)O(\log\log n). Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on δ\delta in our results and the above td(H)O(loglogn)\text{td}(H)\in O(\log\log n) bound are both best possible. We prove that nn-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth tt and a complete graph of size O(n1/t)O(n^{1/t}), which is best possible. Finally, we investigate the conjecture that for any hereditary graph class G\mathcal{G} that admits O(n1ϵ)O(n^{1-\epsilon}) separators, every nn-vertex graph in G\mathcal{G} is a subgraph of the strong product of a graph HH with bounded tree-width and a complete graph of size O(n1ϵ)O(n^{1-\epsilon}). We prove this for various classes G\mathcal{G} of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial Expansion Classes" which had an error, added section "Lower Bounds", and added a new author; v4: minor revisions and corrections

    On generalised majority edge-colourings of graphs

    Full text link
    A 1k\frac{1}{k}-majority ll-edge-colouring of a graph GG is a colouring of its edges with ll colours such that for every colour ii and each vertex vv of GG, at most 1k\frac{1}{k}'th of the edges incident with vv have colour ii. We conjecture that for every integer k2k\geq 2, each graph with minimum degree δk2\delta\geq k^2 is 1k\frac{1}{k}-majority (k+1)(k+1)-edge-colourable and observe that such result would be best possible. This was already known to hold for k=2k=2. We support the conjecture by proving it with 2k22k^2 instead of k2k^2, which confirms the right order of magnitude of the conjectured optimal lower bound for δ\delta. We at the same time improve the previously known bound of order k3logkk^3\log k, based on a straightforward probabilistic approach. As this technique seems not applicable towards any further improvement, we use a more direct non-random approach. We also strengthen our result, in particular substituting 2k22k^2 by (74+o(1))k2(\frac{7}{4}+o(1))k^2. Finally, we provide the proof of the conjecture itself for k4k\leq 4 and completely solve an analogous problem for the family of bipartite graphs.Comment: 18 page

    A proof of the Ryser-Brualdi-Stein conjecture for large even nn

    Full text link
    A Latin square of order nn is an nn by nn grid filled using nn symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every Latin square of order nn contains a transversal with n1n-1 cells, and a transversal with nn cells if nn is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order nn has a transversal with nO(logn/loglogn)n-O(\log n/\log\log n) cells. Here, we show, for sufficiently large nn, that every Latin square of order nn has a transversal with n1n-1 cells. We also apply our methods to show that, for sufficiently large nn, every Steiner triple system of order nn has a matching containing at least (n4)/3(n-4)/3 edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with n/3O(logn/loglogn)n/3-O(\log n/\log\log n) edges, and proves a conjecture of Brouwer from 1981 for large nn.Comment: 71 pages, 13 figure

    HM 32: New Interpretations in Naval History

    Get PDF
    Selected papers from the twenty-first McMullen Naval History Symposium held at the U.S. Naval Academy, 19–20 September 2019.https://digital-commons.usnwc.edu/usnwc-historical-monographs/1031/thumbnail.jp

    Fast Approximation of Search Trees on Trees with Centroid Trees

    Get PDF

    Breaking the All Subsets Barrier for Min k-Cut

    Get PDF

    Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring

    Get PDF
    The Lovász Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have n ‘bad events’, each of which occurs with probability at most p and is independent of all but d other events, then under certain criteria on p and d, all of the bad events can be avoided with positive probability. While the original proof was existential, there has been much study on the algorithmic Lovász Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM ’10] also implied an efficient distributed algorithm for the problem, running in O(log2 n) rounds. For instances with low d, this was improved to O(d 2 + logO(1) log n) by Fischer and Ghaffari [DISC ’17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP ’19]). We give an improved algorithm for the Lovász Local Lemma, providing a trade-off between the strength of the criterion relating p and d, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari’s algorithm, we improve the round complexity to O( d log d + logO(1) log n). At the other end of the trade-off, we obtain a logO(1) log n round complexity for a substantially wider regime than previously known. As our main application, we also give the first logO(1) log n-round distributed algorithm for the problem of ∆+o(∆)-edge coloring a graph of maximum degree ∆. This is an almost exponential improvement over previous results: no prior logo(1) n-round algorithm was known even for 2∆ − 2-edge coloring
    corecore