102 research outputs found
The Moser-Tardos Framework with Partial Resampling
The resampling algorithm of Moser \& Tardos is a powerful approach to develop
constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this
to partial resampling: when a bad event holds, we resample an
appropriately-random subset of the variables that define this event, rather
than the entire set as in Moser & Tardos. This is particularly useful when the
bad events are determined by sums of random variables. This leads to several
improved algorithmic applications in scheduling, graph transversals, packet
routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006)
on graph transversals asymptotically, and obtain improved approximation ratios
for a packet routing problem of Leighton, Maggs, & Rao (1994)
An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles
The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It
gives a sufficient condition on a probability space and a collection of events
for the existence of an outcome that simultaneously avoids all of those events.
Finding such an outcome by an efficient algorithm has been an active research
topic for decades. Breakthrough work of Moser and Tardos (2009) presented an
efficient algorithm for a general setting primarily characterized by a product
structure on the probability space.
In this work we present an efficient algorithm for a much more general
setting. Our main assumption is that there exist certain functions, called
resampling oracles, that can be invoked to address the undesired occurrence of
the events. We show that, in all scenarios to which the original Lovasz Local
Lemma applies, there exist resampling oracles, although they are not
necessarily efficient. Nevertheless, for essentially all known applications of
the Lovasz Local Lemma and its generalizations, we have designed efficient
resampling oracles. As applications of these techniques, we present new results
for packings of Latin transversals, rainbow matchings and rainbow spanning
trees.Comment: 47 page
Algorithmic and enumerative aspects of the Moser-Tardos distribution
Moser & Tardos have developed a powerful algorithmic approach (henceforth
"MT") to the Lovasz Local Lemma (LLL); the basic operation done in MT and its
variants is a search for "bad" events in a current configuration. In the
initial stage of MT, the variables are set independently. We examine the
distributions on these variables which arise during intermediate stages of MT.
We show that these configurations have a more or less "random" form, building
further on the "MT-distribution" concept of Haeupler et al. in understanding
the (intermediate and) output distribution of MT. This has a variety of
algorithmic applications; the most important is that bad events can be found
relatively quickly, improving upon MT across the complexity spectrum: it makes
some polynomial-time algorithms sub-linear (e.g., for Latin transversals, which
are of basic combinatorial interest), gives lower-degree polynomial run-times
in some settings, transforms certain super-polynomial-time algorithms into
polynomial-time ones, and leads to Las Vegas algorithms for some coloring
problems for which only Monte Carlo algorithms were known.
We show that in certain conditions when the LLL condition is violated, a
variant of the MT algorithm can still produce a distribution which avoids most
of the bad events. We show in some cases this MT variant can run faster than
the original MT algorithm itself, and develop the first-known criterion for the
case of the asymmetric LLL. This can be used to find partial Latin transversals
-- improving upon earlier bounds of Stein (1975) -- among other applications.
We furthermore give applications in enumeration, showing that most applications
(where we aim for all or most of the bad events to be avoided) have many more
solutions than known before by proving that the MT-distribution has "large"
min-entropy and hence that its support-size is large
Parallel algorithms and concentration bounds for the Lovasz Local Lemma via witness DAGs
The Lov\'{a}sz Local Lemma (LLL) is a cornerstone principle in the
probabilistic method of combinatorics, and a seminal algorithm of Moser &
Tardos (2010) provides an efficient randomized algorithm to implement it. This
can be parallelized to give an algorithm that uses polynomially many processors
and runs in time on an EREW PRAM, stemming from
adaptive computations of a maximal independent set (MIS). Chung et al. (2014)
developed faster local and parallel algorithms, potentially running in time
, but these algorithms require more stringent conditions than the
LLL.
We give a new parallel algorithm that works under essentially the same
conditions as the original algorithm of Moser & Tardos but uses only a single
MIS computation, thus running in time on an EREW PRAM. This can
be derandomized to give an NC algorithm running in time as well,
speeding up a previous NC LLL algorithm of Chandrasekaran et al. (2013).
We also provide improved and tighter bounds on the run-times of the
sequential and parallel resampling-based algorithms originally developed by
Moser & Tardos. These apply to any problem instance in which the tighter
Shearer LLL criterion is satisfied
Simple Local Computation Algorithms for the General Lovasz Local Lemma
We consider the task of designing Local Computation Algorithms (LCA) for
applications of the Lov\'{a}sz 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~\cite{alon_lll} giving algorithms under significantly stronger LLL-like
conditions. LCAs under those stronger conditions were given in~\cite{Ronitt},
where it was asked if the Moser-Tardos algorithm can be used to design LCAs
under the standard LLL condition. The main contribution of this paper is to
answer this question affirmatively. In fact, our techniques yield LCAs for
settings beyond the standard LLL condition
Deterministic algorithms for the Lovász Local Lemma
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Includes bibliographical references (p. 34-36).The Lovász Local Lemma [6] (LLL) is a powerful result in probability theory that states that the probability that none of a set of bad events happens is nonzero if the probability of each event is small compared to the number of events that depend on it. It is often used in combination with the probabilistic method for non-constructive existence proofs. A prominent application is to k-CNF formulas, where LLL implies that, if every clause in the formula shares variables with at most d < 2k/e other clauses then such the formula has a satisfying assignment. Recently, a randomized algorithm to efficiently construct a satisfying assignment was given by Moser [17]. Subsequently Moser and Tardos [18] gave a randomized algorithm to construct the structures guaranteed by the LLL in a very general algorithmic framework. We address the main problem left open by Moser and Tardos of derandomizing these algorithms efficiently. Specifically, for a k-CNF formula with m clauses and d < 2k/(l+)/e for some c E (0, 1), we give an algorithm that finds a satisfying assignment in time O(m2(1+1/E)). This improves upon the deterministic algorithms of Moser and of Moser- Tardos with running times mn (k2) and mD(k 1/c) which are superpolynomial for k = w(1) and upon other previous algorithms which work only for d </= 2k/ 16 /e. Our algorithm works efficiently for the asymmetric version of LLL under the algorithmic framework of Moser and Tardos [18] and is also parallelizable, i.e., has polylogarithmic running time using polynomially many processors.by Bernhard Haeupler.S.M
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