树上分布式局部可检查问题的复杂度景观

局部可检查标记中局部计算算法的新复杂度类

New Complexity Classes in Locally Checkable Labeling for Local Computation Algorithms

本文聚焦局部计算算法，通过复杂度分类理解LCL问题，在VOLUME模型中给出新的LCL复杂度构造并推广到LCAs，展示了特定探针复杂度的LCLs，采用堆叠实例的方法，区别于相关工作。

论树上局部可检查问题的节点平均复杂度

On the Node-Averaged Complexity of Locally Checkable Problems on Trees

本文研究了树上LCL问题的节点平均复杂度，证明了最坏情况复杂度为$O(\log n)$的问题确定性和随机化平均复杂度分别为$O(\log^* n)$和$O(1)$，并为$Θ(n^{1/k})$问题确定了紧下界。

The Landscape of Distributed Complexities on Trees and Beyond

We study the local complexity landscape of locally checkable labeling (LCL) problems on constant-degree graphs with a focus on complexities below $\log^* n$. Our contribution is threefold: Our main contribution is that we complete the classification of the complexity landscape of LCL problems on trees in the LOCAL model, by proving that every LCL problem with local complexity $o(\log^* n)$ has actually complexity $O(1)$. This result improves upon the previous speedup result from $o(\log \log^* n)$ to $O(1)$ by [Chang, Pettie, FOCS 2017]. In the related LCA and Volume models [Alon, Rubinfeld, V…

Local Mending

In this work we introduce the graph-theoretic notion of mendability: for each locally checkable graph problem we can define its mending radius, which captures the idea of how far one needs to modify a partial solution in order to "patch a hole." We explore how mendability is connected to the existence of efficient algorithms, especially in distributed, parallel, and fault-tolerant settings. It is easy to see that $O(1)$-mendable problems are also solvable in $O(\log^* n)$ rounds in the LOCAL model of distributed computing. One of the surprises is that in paths and cycles, a converse also holds…

分布式问题的自动加速定理

An Automatic Speedup Theorem for Distributed Problems

最近，Brandt等人。[STOC'16]证明了分布式Lovász局部引理的下界，Chang和Pettie [FOCS'17]已经证明了该下界对于充分放松的LLL准则是紧的。他们的结果的核心在于一个加速技术，对于周长至少为$2t+2$的图，将一个特定LLL问题的任何$t$轮算法转换为同一问题的$(t-1)$轮算法。我们大大提高了这种技术，表明这样的加速存在于任何本地可检查的问题$Π$，与不同的是，问题$Π_1$推断$(t-1)$轮算法解决的是不（一定）相同的问题作为$Π$。我们的加速是自动的，因为有一个固定的过程将$Π$的描述转换为$Π_1$的描述，并且是可逆的，因为用于$Π_1$的任何$(t-1)$轮算法都可以转换为用于$Π$的$t$轮算法。特别地，对于具有$n$节点、最大节点度$Δ$和围长至少为$2t+2$的图上的具有精确确定性时间复杂度$T(n, Δ) \leq t$的任何局部可检查问题$Π$，存在具有时间复杂度$T(n, Δ)-1, T(n, Δ)-2, \dots$的问题$Π_1, Π_2, \dots$的序列，其可以从$Π$推断。作为我们的广义加速的第一个应用程序，我们解决了一个长期存在的开放问题Naor和Stockmeyer [STOC'93]：我们表明，弱$2$染色奇数度图不能解决在$o(\log^* Δ)$轮，从而提供了一个匹配的下界，他们的上限。

MPC模型下树中的快速动态规划

Fast Dynamic Programming in Trees in the MPC Model

本研究提出MPC模型下的确定性算法，以O(log D)轮求解树中多种动态规划及图优化等问题，运行时间与节点数无关，严格优于现有两类算法。

What can be sampled locally?

The local computation of Linial [FOCS'87] and Naor and Stockmeyer [STOC'93] concerns with the question of whether a locally definable distributed computing problem can be solved locally: for a given local CSP whether a CSP solution can be constructed by a distributed algorithm using local information. In this paper, we consider the problem of sampling a uniform CSP solution by distributed algorithms, and ask whether a locally definable joint distribution can be sampled from locally. More broadly, we consider sampling from Gibbs distributions induced by weighted local CSPs in the LOCAL model. W…

有界最小宽度图上最长诱导路和诱导不相交路问题的多项式时间算法

Polynomial-time algorithms for the Longest Induced Path and Induced Disjoint Paths problems on graphs of bounded mim-width

我们给出了第一个多项式时间算法的图有界最大诱导匹配宽度（最小宽度）的问题，是不是局部检查。特别是，我们给$n^{\mathcal{O}(w)}$ -时间算法的最小宽度至多$w$，当给定的分解，为以下问题：最长诱导路径，诱导不相交的路径和$H$ -诱导拓扑子固定$H$。我们的结果意味着以下图类具有多项式时间算法这三个问题：区间和双区间图，圆弧，置换和循环置换图，凸图，$k$ -梯形，圆形$k$-梯形，$k$ -多边形，Dilworth- $k$和Co- $k$ -退化图固定$k$。

Improved Deterministic $(Δ+1)$-Coloring in Low-Space MPC

We present a deterministic $O(\log \log \log n)$-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of $(Δ+1)$-coloring on $n$-vertex graphs. In this model, every machine has a sublinear local memory of size $n^ϕ$ for any arbitrary constant $ϕ\in (0,1)$. Our algorithm works under the relaxed setting where each machine is allowed to perform exponential (in $n^ϕ$) local computation, while respecting the $n^ϕ$ space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious $(Δ+1)$-coloring LOCAL algorithm by Chang-L…

Towards a Complexity Classification of LCL Problems in Massively Parallel Computation

In this work, we develop the low-space Massively Parallel Computation (MPC) complexity landscape for a family of fundamental graph problems on trees. We present a general method that solves most locally checkable labeling (LCL) problems exponentially faster in the low-space MPC model than in the LOCAL message passing model. In particular, we show that all solvable LCL problems on trees can be solved in $O(\log n)$ time (high-complexity regime) and that all LCL problems on trees with deterministic complexity $n^{o(1)}$ in the LOCAL model can be solved in $O(\log \log n)$ time (mid-complexity re…

网格上的LCL问题

LCL problems on grids

LCL或局部可检查标记问题（例如最大独立集，最大匹配和顶点着色）在计算的循环（环形1维网格）中得到了很好的理解：每个问题都有$O(1)$，$Θ(\log^* n)$或$Θ(n)$的复杂性，并且最优算法的设计可以完全自动化。本文发展了二维环形网格LCL问题的复杂性理论。复杂度类与一维情况相同：$O(1)$、$Θ(\log^* n)$和$Θ(n)$。然而，给定一个LCL问题，它的复杂性是$Θ(\log^* n)$或$Θ(n)$在二维网格是不可判定的。然而，如果我们正确地猜测问题的复杂性是$Θ(\log^* n)$，我们就可以完全自动化地设计最优算法。对于任何问题，我们可以找到一个算法，是一个范式$A' \circ S_k$，其中$A'$是一个有限的功能，$S_k$是一个算法，以寻找一个最大的独立集的$k$ th权力的网格，和$k$是一个常数。最后，部分自动化设计工具的帮助下，我们分类的复杂性，几个具体的拼箱问题有关的着色和方向。

Lovász Local引理的次对数分布式算法及其复杂性

Sublogarithmic Distributed Algorithms for Lovász Local lemma, and the Complexity Hierarchy

局部可检查标号（LCL）问题基本上包含了$\mathsf{LOCAL}$分布式算法的所有经典问题。在最近的启发性启示中，Chang和Pettie [arXiv 1704.06297]表明，任何具有$o(\log n)$轮随机算法的LCL（在有界度图上）都可以在$T_{LLL}(n)$轮中求解，这是求解有界度$n$节点图上的Lovász局部引理（LLL）（的放松变体）的随机复杂性。目前，关于$T_{LLL}(n)$的最著名的上限是由Chung、Pettie和Su [PODC'14]提出的$O(\log n)$，而最著名的下限是由Brandt等人[STOC'16]提出的$Ω(\log\log n)$。Chang和Pettie指出应该有一个$O(\log\log n)$轮算法。对这一猜想的进展的第一步，并提供了一个显着的改进算法的Chung等。[PODC'14]，我们证明了$T_{LLL}(n)= 2^{O(\sqrt{\log\log n})}$。因此，用于有界度图上的任何LCL问题的任何$o(\log n)$轮随机分布式算法可以自动加速以在$2^{O(\sqrt{\log\log n})}$轮中运行。使用这种改进和一些其他的想法，我们也提高了一些图着色问题的复杂性（在任意度图）从$O(\log n)$轮结果的Chung，Pettie和苏[PODC'14]到$2^{O(\sqrt{\l…

从分布式算法、有限因子与描述性组合数学视角看网格上的局部问题

Local Problems on Grids from the Perspective of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

本研究建立了分布式局部算法、独立同分布过程有限因子与描述性组合数学三个领域的关联，通过研究网格图局部可校验标注，证明有限因子领域的时间层次定理，解答了相关开放问题，推动了统一局部性理论的构建。