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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...
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...
We give an efficient algorithm that, given a graph G and a partition V1,…,V m of its vertex set, finds either an independent transversal (an independent set {v1,…,v m } in G such that ${v_i} \in {V_i}$ for each i), or a subset ${\cal B}$ of vertex classes such that the subgraph of G induced by $\bigcup\nolimits_{\cal B}$ has a small dominating set. A non-algorithmic proof of this result has been known for a number of years and has been used to solve many other problems. Thus we are able to give algorithmic versions of many of these applications, a few of which we describe explicitly here.
The Lovász Local Lemma (LLL) is a probabilistic tool which shows that, if a collection of "bad" events B in a probability space are not too likely and not too interdependent, then there is a positive probability that no bad-events in B occur. Moser & Tardos (2010) gave sequential and parallel algorithms which transformed most applications of the variable-assignment LLL into efficient algorithms. A framework of Harvey & Vondrák (2015) based on "resampling oracles" extended this give very general sequential algorithms for other probability spaces satisfying the Lopsided Lovász Local Lemma (LLLL).We describe a new structural property of resampling oracles which holds for all known resampling oracles, which we call "obliviousness." Essentially, it means that the interaction between two bad-events B, B ′ depends only on the randomness used to resample B, and not on the precise state within B itself.This property has two major consequences. First, it is the key to achieving a unified parallel LLLL algorithm, which is faster than previous, problem-specific algorithms of Harris (2016) for the variable-assignment LLLL algorithm and of Harris & Srinivasan (2014) for permutations. This new algorithm extends a framework of Kolmogorov (2016), and gives the first RNC algorithms for rainbow perfect matchings and rainbow hamiltonian cycles of K n .Second, this property allows us to build LLLL probability spaces out of a relatively simple "atomic" set of events. It was intuitively clear that existing LLLL spaces were built in this way; but the obliviousness property formalizes this and gives a way of automatically turning a resampling oracle for atomic events into a resampling oracle for conjunctions of them. Using this framework, we get the first sequential resampling oracle for rainbow perfect matchings on the complete s-uniform hypergraph K (s) n , and the first commutative resampling oracle for hamiltonian cycles of K 0 perfect matchings of K n [30], perfect matchings of the complete s-uniform hypergraph K (s) n [28], and spanning trees of K n [28].The variable-assignment setting provides one of the simplest forms of the LLLL, and the original algorithm of Moser & Tardos applies to it. In [21], Harris & Srinivasan developed an algorithm similar to the MT algorithm for the probability space of random permutations, which includes the Latin transversal application of [10]. This algorithm was very problem-specific; a more recent line of research has been developing generic LLLL algorithms, which can cover most of the probabilistic forms of the LLLL.Harvey & Vondrák [23] developed a general framework based on a "resampling oracle" R for the probability space. This is a randomized algorithm which, given some state u with some bad-event B true on u, attempts to "rerandomize" the configuration in a minimal way to fix B. This is similar to the way that the MT algorithm rerandomizes the variables involved in a bad-event. Given this resampling oracle, the following Algorithm 2 (which is a generalization of the MT algorithm) can be used...
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