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1. Faster Dynamic-Consistency Checking for Conditional Simple Temporal Networks

   Hunsberger, Luke; Posenato, Roberto (October 2020, 30th International Joint Conference on Applied Planning and Scheduling (ICAPS-2020))

   A Conditional Simple Temporal Network (CSTN) is a structure for representing and reasoning about time in domains where temporal constraints may be conditioned on outcomes of observations made in real time. A CSTN is dynamically consistent (DC) if there is a strategy for executing its timepoints such that all relevant constraints will necessarily be satisfied no matter which outcomes happen to be observed. The literature on CSTNs contains only one sound-and-complete DC-checking algorithm that has been implemented and empirically evaluated. It is a graph-based algorithm that propagates labeled constraints/edges. A second algorithm has been proposed, but not evaluated. It aims to speed up DC checking by more efficiently dealing with so-called negative q-loops. This paper presents a new two-phase approach to DC-checking for CSTNs. The first phase focuses on identifying negative q-loops and labeling key time-points within them. The second phase focuses on computing (labeled) distances from each time-point to a single sink node. The new algorithm, which is also sound and complete for DC-checking, is then empirically evaluated against both pre-existing algorithms and shown to be much faster across not only previously published benchmark problems, but also a new set of benchmark problems. The results show that, on DC instances, the new algorithm tends to be an order of magnitude faster than both existing algorithms. On all other benchmark cases, the new algorithm performs better than or equivalently to the existing algorithms. 

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2. On k-Mer-Based and Maximum Likelihood Estimation Algorithms for Trace Reconstruction

   Cheng, Kuan; Grigorescu, Elena; Li, Xin; Sudan, Madhu; Zhu, Minshen (July 2024, IEEE Press)

   The goal of the trace reconstruction problem is to recover a string x E {0, 1} given many independent traces of x, where a trace is a subsequence obtained from deleting bits of x independently with some given probability. In this paper we consider two kinds of algorithms for the trace reconstruction problem. We first observe that the state-of-the-art result of Chase (STOC 2021), which is based on statistics of arbitrary length-k subsequences, can also be obtained by considering the “k-mer statistics”, i.e., statistics regarding occurrences of contiguous k-bit strings (a.k.a, k-mers) in the initial string x, for k = Mazooji and Shomorony (ISIT 2023) show that such statistics (called k-mer density map) can be estimated within accuracy from poly(n, 2k, l/e) traces. We call an algorithm to be k-mer-based if it reconstructs x given estimates of the k-mer density map. Such algorithms essentially capture all the analyses in the worst-case and smoothed-complexity models of the trace reconstruction problem we know of so far. Our first, and technically more involved, result shows that any k-mer-based algorithm for trace reconstruction must use exp n)) traces, under the assumption that the estimator requires poly(2k, 1 e) traces, thus establishing the optimality of this number of traces. Our analysis also shows that the analysis technique used by Chase is essentially tight, and hence new techniques are needed in order to improve the worst-case upper bound. Our second, simple, result considers the performance of the Maximum Likelihood Estimator (MLE), which specifically picks the source string that has the maximum likelihood to generate the samples (traces). We show that the MLE algorithm uses a nearly optimal number of traces, i.e., up to a factor of n in the number of samples needed for an optimal algorithm, and show that this factor of n loss may be necessary under general “model estimation” settings. 

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3. On k-Mer-Based and Maximum Likelihood Estimation Algorithms for Trace Reconstruction

   Cheng, Kuan; Grigorescu, Elena; Li, Xin; Sudan, Madhu; Zhu, Minshen (July 2024, IEEE Press)

   The goal of the trace reconstruction problem is to recover a string x E {0, 1} given many independent traces of x, where a trace is a subsequence obtained from deleting bits of x independently with some given probability. In this paper we consider two kinds of algorithms for the trace reconstruction problem. We first observe that the state-of-the-art result of Chase (STOC 2021), which is based on statistics of arbitrary length-k subsequences, can also be obtained by considering the “k-mer statistics”, i.e., statistics regarding occurrences of contiguous k-bit strings (a.k.a, k-mers) in the initial string x, for k = Mazooji and Shomorony (ISIT 2023) show that such statistics (called k-mer density map) can be estimated within accuracy from poly(n, 2k, l/e) traces. We call an algorithm to be k-mer-based if it reconstructs x given estimates of the k-mer density map. Such algorithms essentially capture all the analyses in the worst-case and smoothed-complexity models of the trace reconstruction problem we know of so far. Our first, and technically more involved, result shows that any k-mer-based algorithm for trace reconstruction must use exp n)) traces, under the assumption that the estimator requires poly(2k, 1 e) traces, thus establishing the optimality of this number of traces. Our analysis also shows that the analysis technique used by Chase is essentially tight, and hence new techniques are needed in order to improve the worst-case upper bound. Our second, simple, result considers the performance of the Maximum Likelihood Estimator (MLE), which specifically picks the source string that has the maximum likelihood to generate the samples (traces). We show that the MLE algorithm uses a nearly optimal number of traces, i.e., up to a factor of n in the number of samples needed for an optimal algorithm, and show that this factor of n loss may be necessary under general “model estimation” settings. 

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4. Fast Optimal Circular Clustering and Applications on Round Genomes

   https://doi.org/10.1109/TCBB.2021.3077573  

   Debnath, Tathagata; Song, Mingzhou (June 2021, IEEE/ACM Transactions on Computational Biology and Bioinformatics)

   null (Ed.)

   Round genomes are found in bacteria, plant chloroplasts, and mitochondria. Genetic or epigenetic marks can present biologically interesting clusters along a circular genome. The circular data clustering problem groups N points on a circle into K clusters to minimize the within-cluster sum of squared distances. Repeatedly applying the K-means algorithm takes quadratic time, impractical for large circular datasets. To overcome this issue, we developed a fast, reproducible, and optimal circular clustering (FOCC) algorithm of worst-case O(KN log^2 N) time. The core is a fast optimal framed clustering algorithm, which we designed by integrating two divide-and-conquer and one bracket dynamic programming strategies. The algorithm is optimal based on a property of monotonic increasing cluster borders over frames on linearized data. On clustering 50,000 circular data points, FOCC outruns brute-force or heuristic circular clustering by three orders of magnitude. We produced clusters of CpG sites and genes along three round genomes, exhibiting higher quality than heuristic clustering. More broadly, the presented subquadratic-time algorithms offer the fastest known solution to not only framed and circular clustering, but also angular, periodical, and looped clustering. We implemented these algorithms in the R package OptCirClust (https://CRAN.R-project.org/package=OptCirClust) 

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5. On Super Strong ETH

   https://doi.org/10.1007/978-3-030-24258-9_28  

   Vyas, N.; Williams, R. (January 2019, Theory and Applications of Satisfiability Testing – SAT 2019)

   Multiple known algorithmic paradigms (backtracking, local search and the polynomial method) only yield a 2n(1−1/O(k)) time algorithm for k-SAT in the worst case. For this reason, it has been hypothesized that the worst-case k-SAT problem cannot be solved in 2n(1−f(k)/k) time for any unbounded function f. This hypothesis has been called the “Super-Strong ETH”, modeled after the ETH and the Strong ETH. We give two results on the Super-Strong ETH: 1. It has also been hypothesized that k-SAT is hard to solve for randomly chosen instances near the “critical threshold”, where the clause-to-variable ratio is 2^kln2−Θ(1). We give a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT and planted k-SAT for any clause-to-variable ratio. For example, given any random k-SAT instance F with n variables and m clauses, our algorithm decides satisfiability for F in 2^n(1−Ω(logk)/k) time, with high probability (over the choice of the formula and the randomness of the algorithm). It turns out that a well-known algorithm from the literature on SAT algorithms does the job: the PPZ algorithm of Paturi, Pudlák and Zane [17]. 2. The Unique k-SAT problem is the special case where there is at most one satisfying assignment. Improving prior reductions, we show that the Super-Strong ETHs for Unique k-SAT and k-SAT are equivalent. More precisely, we show the time complexities of Unique k-SAT and k-SAT are very tightly correlated: if Unique k-SAT is in 2^n(1−f(k)/k) time for an unbounded f, then k-SAT is in 2^n(1−f(k)(1−ε)/k) time for every ε>0. 

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