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1. 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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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. Trace Reconstruction from Local Statistical Queries

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.52  

   Chen, Xi; De, Anindya; Lee, Chin Ho; Servedio, Rocco A (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Kumar, Amit; Ron-Zewi, Noga (Ed.)

   The goal of trace reconstruction is to reconstruct an unknown n-bit string x given only independent random traces of x, where a random trace of x is obtained by passing x through a deletion channel. A Statistical Query (SQ) algorithm for trace reconstruction is an algorithm which can only access statistical information about the distribution of random traces of x rather than individual traces themselves. Such an algorithm is said to be 𝓁-local if each of its statistical queries corresponds to an 𝓁-junta function over some block of 𝓁 consecutive bits in the trace. Since several - but not all - known algorithms for trace reconstruction fall under the local statistical query paradigm, it is interesting to understand the abilities and limitations of local SQ algorithms for trace reconstruction. In this paper we establish nearly-matching upper and lower bounds on local Statistical Query algorithms for both worst-case and average-case trace reconstruction. For the worst-case problem, we show that there is an Õ(n^{1/5})-local SQ algorithm that makes all its queries with tolerance τ ≥ 2^{-Õ(n^{1/5})}, and also that any Õ(n^{1/5})-local SQ algorithm must make some query with tolerance τ ≤ 2^{-Ω̃(n^{1/5})}. For the average-case problem, we show that there is an O(log n)-local SQ algorithm that makes all its queries with tolerance τ ≥ 1/poly(n), and also that any O(log n)-local SQ algorithm must make some query with tolerance τ ≤ 1/poly(n). 

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4. Triplet Reconstruction and all other Phylogenetic CSPs are Approximation Resistant

   https://doi.org/10.1109/FOCS57990.2023.00024  

   Chatziafratis, Vaggos; Makarychev, Konstantin (November 2023, IEEE)

   We study the natural problem of Triplet Reconstruction (also known as Rooted Triplets Consistency or Triplet Clustering), originally motivated by applications in computational biology and relational databases (Aho, Sagiv, Szymanski, and Ullman, 1981) [2]: given n datapoints, we want to embed them onto the n leaves of a rooted binary tree (also known as a hierarchical clustering, or ultrametric embedding) such that a given set of m triplet constraints is satisfied. A triplet constraint i j · k for points i, j, k indicates that 'i, j are more closely related to each other than to k,' (in terms of distances d(i, j) ≤ d(i, k) and d(i, j) ≤ d(j, k)) and we say that a tree satisfies the triplet i j · k if the distance in the tree between i, j is smaller than the distance between i, k (or j, k). Among all possible trees with n leaves, can we efficiently find one that satisfies a large fraction of the m given triplets? Aho et al. (1981) [2] studied the decision version and gave an elegant polynomial-time algorithm that determines whether or not there exists a tree that satisfies all of the m constraints. Moreover, it is straightforward to see that a random binary tree achieves a constant 13-approximation, since there are only 3 distinct triplets i j|k, i k| j, j k · i (each will be satisfied w.p. 13). Unfortunately, despite more than four decades of research by various communities, there is no better approximation algorithm for this basic Triplet Reconstruction problem.Our main theorem-which captures Triplet Reconstruction as a special case-is a general hardness of approximation result about Constraint Satisfaction Problems (CSPs) over infinite domains (CSPs where instead of boolean values {0,1} or a fixed-size domain, the variables can be mapped to any of the n leaves of a tree). Specifically, we prove that assuming the Unique Games Conjecture [57], Triplet Reconstruction and more generally, every Constraint Satisfaction Problem (CSP) over hierarchies is approximation resistant, i.e., there is no polynomial-time algorithm that does asymptotically better than a biased random assignment.Our result settles the approximability not only for Triplet Reconstruction, but for many interesting problems that have been studied by various scientific communities such as the popular Quartet Reconstruction and Subtree/Supertree Aggregation Problems. More broadly, our result significantly extends the list of approximation resistant predicates by pointing to a large new family of hard problems over hierarchies. Our main theorem is a generalization of Guruswami, Håstad, Manokaran, Raghavendra, and Charikar (2011) [36], who showed that ordering CSPs (CSPs over permutations of n elements, e.g., Max Acyclic Subgraph, Betweenness, Non-Betweenness) are approximation resistant. The main challenge in our analyses stems from the fact that trees have topology (in contrast to permutations and ordering CSPs) and it is the tree topology that determines whether a given constraint on the variables is satisfied or not. As a byproduct, we also present some of the first CSPs where their approximation resistance is proved against biased random assignments, instead of uniformly random assignments. 

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5. Algorithms for reconstruction over single and multiple deletion channels

   Sundara Rajan Srinivasavaradhan, Michelle Du (June 2021, IEEE transactions on information theory)

   Recent advances in DNA sequencing technology and DNA storage systems have rekindled the interest in deletion channels. Multiple recent works have looked at variants of sequence reconstruction over a single and over multiple deletion channels, a notoriously difficult problem due to its highly combinatorial nature. Although works in theoretical computer science have provided algorithms which guarantee perfect reconstruction with multiple independent observations from the deletion channel, they are only applicable in the large blocklength regime and more restrictively, when the number of observations is also large. Indeed, with only a few observations, perfect reconstruction of the input sequence may not even be possible in most cases. In such situations, maximum likelihood (ML) and maximum aposteriori (MAP) estimates for the deletion channels are natural questions that arise and these have remained open to the best of our knowledge. In this work, we take steps to answer the two aforementioned questions. Specifically: 1. We show that solving for the ML estimate over the single deletion channel (which can be cast as a discrete optimization problem) is equivalent to solving its relaxation, a continuous optimization problem; 2. We exactly compute the symbolwise posterior distributions (under some assumptions on the priors) for both the single as well as multiple deletion channels. As part of our contributions, we also introduce tools to visualize and analyze error events, which we believe could be useful in other related problems concerning deletion channels. 

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