The Lovasz Local Lemma [EL75] is a powerful tool to prove the existence of combinatorial objects meeting a prescribed collection of criteria. The technique can directly be applied to the satisfiability problem, yielding that a k-CNF formula in which each clause has common variables with at most 2^(k-2) other clauses is always satisfiable. All hitherto known proofs of the Local Lemma are non-constructive and do thus not provide a recipe as to how a satisfying assignment to such a formula can be efficiently found. In his breakthrough paper [Bec91], Beck demonstrated that if the neighbourhood of each clause be restricted to O(2^(k/48)), a polynomial time algorithm for the search problem exists. Alon simplified and randomized his procedure and improved the bound to O(2^(k/8)) [Alo91]. Srinivasan presented in [Sri08] a variant that achieves a bound of essentially O(2^(k/4)). In [Mos08], we improved this to O(2^(k/2)). In the present paper, we give a randomized algorithm that finds a satisfying assignment to every k-CNF formula in which each clause has a neighbourhood of at most the asymptotic optimum of 2^(k-5)-1 other clauses and that runs in expected time polynomial in the size of the formula, irrespective of k. If k is considered a constant, we can also give a deterministic variant. In contrast to all previous approaches, our analysis does not anymore invoke the standard non-constructive versions of the Local Lemma and can therefore be considered an alternative, constructive proof of it.
We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lovász Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al.~\cite{Ronitt} that have received a lot of attention in recent years. The LLL is an existential, sufficient condition for a collection of sets to have non-empty intersection (in applications, often, each set comprises all objects having a certain property). The ground-breaking algorithm of Moser and Tardos~\cite{MT} made the LLL fully constructive, following earlier results by Beck~\cite{beck_lll} and Alon~\cit…
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 o…
INSAIT, Sofia University “St.˜Kliment Ohridski” and ETH Zürich（INSAIT，索非亚大学“圣·克利门特·奥赫里德斯基”和苏黎世联邦理工学院）；University of California, Davis（加州大学戴维斯分校）；Princeton University（普林斯顿大学）
We consider recent formulations of the algorithmic Lovasz Local Lemma by Achlioptas-Iliopoulos-Kolmogorov [2] and by Achlioptas-Iliopoulos-Sinclair [3]. These papers analyze a random walk algorithm for finding objects that avoid undesired "bad events" (or "flaws"), and prove that under certain conditions the algorithm is guaranteed to find a "flawless" object quickly. We show that conditions proposed in these papers are incomparable, and introduce a new family of conditions that includes those in [2, 3] as special cases. We also consider another condition that appeared in [3] in the context of…
The Local Computation Algorithm (LCA) model is a popular model in the field of sublinear-time algorithms that measures the complexity of an algorithm by the number of probes the algorithm makes in the neighborhood of one node to determine that node's output. In this paper we show that the randomized LCA complexity of the Lovász Local Lemma (LLL) on constant degree graphs is $Θ(\log n)$. The lower bound follows by proving an $Ω(\log n)$ lower bound for the Sinkless Orientation problem introduced in [Brandt et al. STOC 2016]. This answers a question of [Rosenbaum, Suomela PODC 2020]. Additionall…
We give a fast algorithm for sampling uniform solutions of general constraint satisfaction problems (CSPs) in a local lemma regime. Suppose that the CSP has $n$ variables with domain size at most q, each constraint contains at most k variables, shares variables with at most $Δ$ constraints, and is violated with probability at most $p$ by a uniform random assignment. The algorithm returns an almost uniform satisfying assignment in expected $\mathrm{poly}(q,k,Δ)\cdot\tilde{O}(n)$ time, as long as a local lemma condition is satisfied: \[ k\cdot p\cdot q^2\cdot Δ^5\le C_0\quad\text{for a suitably …
We give a Markov chain based perfect sampler for uniform sampling solutions of constraint satisfaction problems (CSP). Under some mild Lovász local lemma conditions where each constraint of the CSP has a small number of forbidden local configurations, our algorithm is accurate and efficient: it outputs a perfect uniform random solution and its expected running time is quasilinear in the number of variables. Prior to our work, perfect samplers are only shown to exist for CSPs under much more restrictive conditions (Guo, Jerrum, and Liu, JACM'19). Our algorithm has two components: 1. A simple pe…
Following the groundbreaking algorithm of Moser and Tardos for the Lovasz Local Lemma (LLL), there has been a plethora of results analyzing local search algorithms for various constraint satisfaction problems. The algorithms considered fall into two broad categories: resampling algorithms, analyzed via different algorithmic LLL conditions; and backtracking algorithms, analyzed via entropy compression arguments. This paper introduces a new convergence condition that seamlessly handles resampling, backtracking, and hybrid algorithms, i.e., algorithms that perform both resampling and backtracking…
We give a simple combinatorial algorithm to deterministically approximately count the number of satisfying assignments of general constraint satisfaction problems (CSPs). Suppose that the CSP has domain size $q=O(1)$, each constraint contains at most $k=O(1)$ variables, shares variables with at most $Δ=O(1)$ constraints, and is violated with probability at most $p$ by a uniform random assignment. The algorithm returns in polynomial time in an improved local lemma regime: \[ q^2\cdot k\cdot p\cdotΔ^5\le C_0\quad\text{for a suitably small absolute constant }C_0. \] Here the key term $Δ^5$ improv…
Recently, Brandt, Maus and Uitto [PODC'19] showed that, in a restricted setting, the dependency of the complexity of the distributed Lovász Local Lemma (LLL) on the chosen LLL criterion exhibits a sharp threshold phenomenon: They proved that, under the LLL criterion $p2^d < 1$, if each random variable affects at most $3$ events, the deterministic complexity of the LLL in the LOCAL model is $O(d^2 + \log^* n)$. In stark contrast, under the criterion $p2^d \leq 1$, there is a randomized lower bound of $Ω(\log \log n)$ by Brandt et al. [STOC'16] and a deterministic lower bound of $Ω(\log n)$ by C…
We study the problem of sampling an approximately uniformly random satisfying assignment for atomic constraint satisfaction problems i.e. where each constraint is violated by only one assignment to its variables. Let $p$ denote the maximum probability of violation of any constraint and let $Δ$ denote the maximum degree of the line graph of the constraints. Our main result is a nearly-linear (in the number of variables) time algorithm for this problem, which is valid in a Lovász local lemma type regime that is considerably less restrictive compared to previous works. In particular, we provide s…
Let $Φ= (V, \mathcal{C})$ be a constraint satisfaction problem on variables $v_1,\dots, v_n$ such that each constraint depends on at most $k$ variables and such that each variable assumes values in an alphabet of size at most $[q]$. Suppose that each constraint shares variables with at most $Δ$ constraints and that each constraint is violated with probability at most $p$ (under the product measure on its variables). We show that for $k, q = O(1)$, there is a deterministic, polynomial time algorithm to approximately count the number of satisfying assignments and a randomized, polynomial time al…
Fast sampling and counting k-SAT solutions in the local lemma regime
我们给出了新的算法基于马尔可夫链的采样和近似计数满足$k$ -一致CNF公式，其中每个变量出现在最多$d$倍的分配。对于满足$kd<n^{o(1)}$和$k\ge 20\log k + 20\log d + 60$的任何$k$和$d$，新的采样算法以接近线性时间运行，并且计数算法以接近二次时间运行。我们的方法受到Moitra（JACM，2019）的启发，Moitra在近似计数中显著地利用了Lovász局部引理。我们的主要技术贡献是使用本地引理绕过传统的马尔可夫链方法中的连接障碍，这使得发达的MCMC方法适用于不连通的状态空间，如SAT解决方案。我们的方法的好处是避免了局部结构的枚举，并获得固定的多项式运行时间，即使$k=ω(1)$或$d=ω(1)$。
We give a Markov chain based algorithm for sampling almost uniform solutions of constraint satisfaction problems (CSPs). Assuming a canonical setting for the Lovász local lemma, where each constraint is violated by a small number of forbidden local configurations, our sampling algorithm is accurate in a local lemma regime, and the running time is a fixed polynomial whose dependency on $n$ is close to linear, where $n$ is the number of variables. Our main approach is a new technique called state compression, which generalizes the "mark/unmark" paradigm of Moitra (Moitra, JACM, 2019), and can gi…
We give polynomial time algorithms for the seminal results of Kahn, who showed that the Goldberg-Seymour and List-Coloring conjectures for (list-)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilistic method and are non-constructive. Our key insight is to show that the main result of Achlioptas, Iliopoulos and Kolmogorov for analyzing local search algorithms can be used to make constructive applications of a powerful version of the so-called Lopsided Lovasz Local Lemma. In particular, we use it to design algorithms that exploit the fact that correlations…
Learning CNF formulas from uniform random solutions in the local lemma regime
School of Computing and Data Science, The University of Hong Kong（计算与数据科学学院，香港大学）；Department of Computer Science, University of California, Santa Barbara（计算机科学系，加州大学圣芭芭拉分校）；State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University（新型软件技术国家重点实验室，南京大学）
