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.
Our approach can be roughly described as forward-looking analysis with a careful modification of the Moser-Tardos algorithm, formulated in the framework of . Our main conceptual contribution is the simple definition of the , which allows the resamplings to be readily incorporated into the forward-looking analysis. Our modification of the Moser-Tardos algorithm is designed to combine this analysis with the technology of stable set sequences'' @cite_9 , defined in stable-set-sequences , which allows us to accommodate various LLL criteria, including Shearer's criterion. This plays a fundamental role in the full proof of LLL-tight-result .
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The resampling algorithm of Moser & Tardos is a powerful approach to develop constructive versions of the Lovasz 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 improve the approximation ratio of a generalized @math -dimensional scheduling problem studied by Azar & Epstein from @math to @math , and settle a conjecture of Szabo & Tardos on graph transversals asymptotically. As a point of comparison with the MT algorithm, we show a family of constraint satisfaction problems that does not have "locality" (every constraint is affected by every variable), and so the LLL and the Moser-Tardos algorithm do not give any useful results; however, our method gives strong scale-free approximation ratios and terminates in expected polynomial time for this family. Beck's early work [3] gave an efficient version of the Lovasz Local Lemma(LLL) with significant compromise in the parameters. Following several improvements [1,7,4,13], Moser [8], and Moser and Tardos [9] obtained asymptotically optimal results in terms of the maximal degree. For a fixed dependency graph G the exact criterion under which LLL applies is given by Shearer in [12]. For a dependency structure G, let LO(G) be the set of those probability assignments to the nodes of G for which the Lovasz Local Lemma holds. We show that: Both the sequential and parallel ersions of the Moser-Tardos algorithm are efficient up to the Shearer's bound, by giving a tighter analysis. We also prove that, whenever p ∈ LO(G) (1+e), the expected running times of the sequential and parallel versions are at most n e and O(1 e log n e), the later when e 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 @math time on an EREW PRAM, stemming from @math adaptive computations of a maximal independent set (MIS). (2014) developed faster local and parallel algorithms, potentially running in time @math , 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 @math time on an EREW PRAM. This can be derandomized to give an NC algorithm running in time @math as well, speeding up a previous NC LLL algorithm of (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. This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations.