Commutativity in the Algorithmic Lovász Local Lemma

Kolmogorov, Vladimir

doi:10.1109/focs.2016.88

Cited by 25 publications

(79 citation statements)

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“…For example, one main motivation of [1,20] was to generalize the Swapping Algorithm. Then, Kolmogorov noted that our Swapping Algorithm had the nice property that the choice of which bad event to resample can be made arbitrarily, a property missing from analysis of [1]; this led to Kolmogorov's work [26] where he partially generalized that property (to which he refers as commutativity).…”

Section: Resultsmentioning

confidence: 99%

“…Section 6 puts the analyses of Sections 3, 4, 5 together, to prove that our Swapping Algorithm terminates in polynomial time under the same conditions as those of Theorem 1.1; also, as mentioned in Section 1.2, Section 6.3 discusses certain contributions that our approach leads to that do not appear to follow from [1,20,26].…”

Section: Outlinementioning

confidence: 99%

“…As one example of the power of our proof method, we develop a parallel Swapping Algorithm in Section 7; we emphasize that such a parallel algorithm cannot be shown using the results of [1] or [20]. A second example is provided by Theorem 8.2, which we do not see how to develop using the frameworks of [1,20,26].…”

Section: Comparison With Other Llll Algorithmsmentioning

confidence: 99%

“…In [1], Achlioptas & Iliopoulos gave an algorithm which is based on a compression analysis for a random walk; this was improved for permutations and matchings by Kolmogorov [26]. In [20], Harvey & Vondrák gave a probabilistic analysis similar to the parallel Moser-Tardos algorithm.…”

Section: Comparison With Other Llll Algorithmsmentioning

confidence: 99%

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Harris¹,

Srinivasan²

2017

Theory of Comput.

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Section: Resultsmentioning

confidence: 99%

Section: Outlinementioning

confidence: 99%

Section: Comparison With Other Llll Algorithmsmentioning

confidence: 99%

Section: Comparison With Other Llll Algorithmsmentioning

confidence: 99%

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Harris¹,

Srinivasan²

2017

Theory of Comput.

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“…Finally, as we will see, we will need to show that the algorithm of Theorem 1.3 is commutative, a notion introduced by Kolmogorov [22]. This fact may be of independent interest since, as shown in [22,14], commutative algorithms have several nice properties: they are typically parallelizable, their output distribution has high entropy, etc.As a final remark, we note that, to the best of our knowledge, Theorem 1.4 is the first result to give an asymptotically optimal polynomial time algorithm for list-edge coloring multigraphs.…”

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Efficiently list-edge coloring multigraphs asymptotically optimally

Iliopoulos

Sinclair

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We give polynomial time algorithms for the seminal results of Kahn [18,19], 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 [2] for analyzing local search algorithms can be used to make constructive applications of a powerful version of the so-called Lopsided Lovász Local Lemma. In particular, we use it to design algorithms that exploit the fact that correlations in the probability spaces on matchings used by Kahn decay with distance. algorithms and techniques are very different from those of [33,35]. Finally, as we will see, we will need to show that the algorithm of Theorem 1.3 is commutative, a notion introduced by Kolmogorov [22]. This fact may be of independent interest since, as shown in [22,14], commutative algorithms have several nice properties: they are typically parallelizable, their output distribution has high entropy, etc.As a final remark, we note that, to the best of our knowledge, Theorem 1.4 is the first result to give an asymptotically optimal polynomial time algorithm for list-edge coloring multigraphs. Technical OverviewThe proofs of Theorems 1.1 and 1.2 are based on a very sophisticated variation of what is known as the semi-random method (also known as the "naive coloring procedure"), which is the main technical tool behind some of the strongest graph coloring results, e.g., [16,17,21,25]. The idea is to gradually color the graph in iterations, until we reach a point where we can finish the coloring using a greedy algorithm. In its most basic form, each iteration consists of the following simple procedure (using vertex coloring as a canonical example): Assign to each vertex a color chosen uniformly at random; then uncolor any vertex which receives the same color as one of its neighbors. Using the Lovász Local Lemma (LLL) [9] and concentration inequalities, one typically shows that, with positive probability, the resulting partial proper coloring has useful properties that allow for the continuation of the argument in the next iteration. For a nice exposition of both the method and the proofs of Theorems 1.1 and 1.2, the reader is referred to [26].The key new ingredient in Kahn's arguments is the method of assigning colors. For each color c, we choose a matching M c from some hard-core distribution on M(G) and assign the color c to the edges in M c . The idea is that by assigning each color exclusively to the edges of one matching, we avoid conflicting color assignments and the resulting uncolorings.The existence of such hard-core distributions is guaranteed by the characterization of the matching polytope due to Edmonds [8] and a result by Lee [23] (also shown independently by Rabinovich et al. [32]). The crucial fact about them is that they are endowed with very useful approximate stochastic independence properties, as was shown by Kahn and Kayll ...

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New bounds for the Moser‐Tardos distribution

Harris

2020

Random Struct Algorithms

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The Lovász Local Lemma (LLL) is a probabilistic principle which has been used in a variety of combinatorial constructions to show the existence of structures that have good "local" properties. In many cases, one wants more information about these structures, other than that they exist. In such case, using the "LLL-distribution", one can show that the resulting combinatorial structures have good global properties in expectation.While the LLL in its classical form is a general existential statement about probability spaces, nearly all applications in combinatorics have been turned into efficient algorithms. The simplest, variable-based setting of the LLL was covered by the seminal algorithm of Moser & Tardos (2010). This was extended by Harris & Srinivasan (2014) to random permutations, and more recently by Achlioptas & Ilioupoulos (2014) and Harvey & Vondrák (2015) to general probability spaces.One can similarly define for these algorithms an "MT-distribution," which is the distribution at the termination of the Moser-Tardos algorithm. Haeupler et al. (2011) showed bounds on the MT-distribution which essentially match the LLL-distribution for the variable-assignment setting; Harris & Srinivasan showed similar results for the permutation setting.In this work, we show new bounds on the MT-distribution which are significantly stronger than those known to hold for the LLL-distribution. In the variable-assignment setting, we show a tighter bound on the probability of a disjunctive event or singleton event. As a consequence, in k-SAT instances with bounded variable occurrence, the MT-distribution satisfies an -approximate independence condition asymptotically stronger than the LLL-distribution. We use this to show a nearly tight bound on the minimum implicate size of a CNF boolean formula. Another noteworthy application is constructing independent transversals which avoid a given subset of vertices; this provides a constructive analogue to a result of Rabern (2014).In the permutation LLL setting, we show a new type of bound which is similar to the clusterexpansion LLL criterion of Bissacot et al. (2011), but is stronger and takes advantage of the extra structure in permutations. We illustrate by showing new, stronger bounds on low-weight Latin transversals and partial Latin transversals.The Lopsided Lovász Local Lemma. Although the variable-assignment LLL is by far the most common setting in combinatorics, there are other probability spaces for which a generalized form of the LLL, known as the Lopsided Lovász Local Lemma or LLLL, applies. This was introduced by Erdős & Spencer [6], which showed that a form of the LLLL applies to the probability space defined by the uniform distribution on permutations of n letters. Erdős & Spencer used this to construct Latin transversals; the space of random permutations has since been used in a variety of other combinatorial constructions. While the space of random permutations (which we refer to as the permutation LLL) is the most well-known application of the LLLL, it also covers p...

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Commutativity in the Algorithmic Lovász Local Lemma

Cited by 25 publications

References 23 publications

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Efficiently list-edge coloring multigraphs asymptotically optimally

New bounds for the Moser‐Tardos distribution

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