- Introduction. The minimum flow decomposition (MFD) problem asks to decompose a directed acyclic flow network (G,f) with a unique source s and a unique sink t into the fewest weighted s-t paths whose combined contributions exactly reproduce f. MFD underlies a broad class of multi-assembly tasks in bioinformatics: reference-based RNA assembly from splice graphs [Trapnell et al., 2010; Guttman et al., 2010; Tomescu et al., 2013; Song et al., 2016; Liu et al., 2016; Pertea et al., 2015; Kovaka et al., 2019; Shao and Kingsford, 2017; Zhang et al., 2022; Tung et al., 2019], metagenomic assembly [Shaw et al., 2024], and viral quasi-species inference [Baaijens et al., 2020]. MFD is strongly NP-hard [Vatinlen et al., 2008] and hard to approximate within some fixed constant factor [Hartman et al., 2012]. Exact solvers include an FPT algorithm whose runtime grows exponentially in the solution size [Kloster et al., 2018] and a family of integer linear programming (ILP) formulations [Dias et al., 2022; Grigorjew et al., 2024] capable of handling extensions such as inexact flows [Williams et al., 2019; Dias and Tomescu, 2024], safety and subpath constraints [Williams et al., 2022; Gibney et al., 2022; Khan et al., 2022; Dias et al., 2023], and graphs with cycles [Dias et al., 2025]. However, ILP remains unscalable on large practical instances. The widely used greedy-width heuristic [Vatinlen et al., 2008] is very efficient but can be exponentially worse than optimal in the worst case [Cáceres et al., 2024]. The state-of-the-art heuristic, catfish [Shao and Kingsford, 2017], substantially improves this efficiency-performance tradeoff by identifying linear equations among edge flow values - structural constraints implied by any optimal decomposition - and resolving them via safe graph transformations. On simpler instances catfish is highly effective, but three interrelated limitations degrade its performance on complex graphs: (1) it cannot distinguish good equations (arising from a true optimal decomposition) from superficial ones that distort the graph when resolved; (2) many good equations cannot be fully resolved due to the absence of suitable closed subgraphs, so catfish discards their information entirely; and (3) when no equation is resolvable catfish falls back to greedy-width, which performs poorly on entangled graphs. - Method. We introduce catfish-LP, which augments catfish with a lightweight linear programming (LP) formulation based on saturating subflows. For each edge e ∈ E we define continuous variables {x_e(a) : a ∈ E} modeling a valid s-t subflow that saturates e; intuitively, x_e(a) represents the amount of flow on e that must passes through edge a. Five base constraints enforce saturation, symmetry, flow validity, and flow conservation. Two additional equation constraints require that the aggregate subflow through the left-hand-side edges of an equation equals that through the right-hand-side edges. The full LP is polynomial-time solvable, adding only modest overhead over catfish. The LP plays three complementary roles within a single unified framework: (1) equation filtering: if adding a candidate equation renders the LP infeasible, that equation cannot arise from any minimum decomposition and is discarded, preventing structurally invalid graph transformations; (2) safe edge merging: a feasible LP solution reveals pairs of edges that must carry identical subflow and can therefore be safely contracted; (3) informed greedy extraction: when no further simplification is possible, rather than invoking greedy-width blindly, catfish-LP extracts from the LP solution the heaviest simple path consistent with all surviving equations, deferring the error-prone greedy step as long as possible. - Experimental Results. We compare catfish-LP against greedy-width, catfish, and the optimized ILP solver [Grigorjew et al., 2024] on two benchmarks, using Gurobi [{Gurobi Optimization, 2024] as the underlying LP/ILP engine. Table 1 reports decomposition quality on four datasets of biologically derived splice graphs, with abundances estimated by Salmon [Patro et al., 2017] or simulated with the Flux-Simulator [Griebel et al., 2012]. Catfish-LP achieves the smallest excess among heuristics on the Salmon dataset and negative excess on the remaining three, matching ILP quality while being orders of magnitude faster. Figure 1 summarizes results on 1,440 simulated graphs spanning 72 complexity configurations. Catfish-LP consistently produces the smallest decompositions and recovers the most ground-truth paths among all heuristics, achieving near-ILP quality in a fraction of its runtime - including a threefold improvement over catfish on the hardest 498 instances where ILP times out on every instance. - Conclusion. Catfish-LP demonstrates that incorporating a polynomial-time LP oracle into a combinatorial heuristic yields substantial gains in decomposition quality with negligible scalability cost, addressing each of catfish’s core limitations in a unified manner. Future directions include strengthened LP formulations, domain-specific constraints for transcriptome assembly, and probabilistic interpretations of LP-guided decompositions.

Abstract Given a set P of n points in $$\mathbb {R}^d$$ R d , and a positive integer $$k \le n$$ k ≤ n , the k -dispersion problem is that of selecting k of the given points so that the minimum inter-point distance among them is maximized (under Euclidean distances). Among others, we show the following: Given a set P of n points in the plane, and a positive integer $$k \ge 2$$ k ≥ 2 , the k -dispersion problem can be solved by an algorithm running in $$O\left( n^{k-1} \log {n}\right) $$ O n k - 1 log n time. This extends an earlier result for $$k=3$$ k = 3 , due to Horiyama, Nakano, Saitoh, Suetsugu, Suzuki, Uehara, Uno, and Wasa [20] to arbitrary k . In particular, it improves on previous running times for small k . Given a set P of n points in $$\mathbb {R}^3$$ R 3 , and a positive integer $$k \ge 2$$ k ≥ 2 , the k -dispersion problem can be solved by an algorithm running in $$ {\left\{ \begin{array}{ll} O\left( n^{k-1} \log {n}\right) \text {time}, & \text {if } k \text { is even};\\ O\left( n^{k-1} \log ^2{n}\right) \text {time}, & \text {if } k \text { is odd}. \end{array}\right. } $$ O n k - 1 log n time , if k is even ; O

MOTIVATION: Seeding is an essential preparatory step for many fundamental computational tasks that require large-scale sequence comparison. Substring-based seeding methods such as kmers are ideal for sequences with low error rates but struggle to achieve high sensitivity while maintaining a reasonable precision for error-prone long reads. SubseqHash, a novel subsequence-based seeding method we recently developed, achieves superior accuracy to substring-based methods in seeding sequences with high mutation/error rates, while the only drawback is its computation speed. RESULTS: We propose SubseqHash2, an improved algorithm that can compute multiple sets of seeds in one run, by defining k orders over all length-k subsequences and finding the optimal subsequence under each of the k orders in a single dynamic programming framework. The algorithm is further accelerated using single instruction, multiple data instructions for parallel computing. The design of SubseqHash2 also allows it to generate the same sets of seeds for a string and its reverse complement by using symmetric random tables. We demonstrate that SubseqHash2 drastically outperforms popular substring-based methods including kmers, minimizers, syncmers, and Strobemers for three fundamental applications. In read mapping, SubseqHash2 can generate adequate seed matches for aligning hard reads that minimap2 fails on. In sequence alignment, SubseqHash2 achieves high coverage of correct seeds and low coverage of incorrect seeds. In overlap detection, seeds produced by SubseqHash2 lead to more correct overlapping pairs at the same false-positive rate. In all experiments, SubseqHash2 achieves a 10-50× speedup over SubseqHash while maintaining nearly identical high accuracy. With all the algorithmic breakthroughs of SubseqHash2, we clear the path for the wide adoption of subsequence-based seeds in long-read analysis. AVAILABILITY AND IMPLEMENTATION: SubseqHash2 is available at https://github.com/Shao-Group/SubseqHash2 and have also been archived on Software Heritage (swh:1:dir:86738fc4b919eb6a9a26f7f533c25eb69f9a96d5).

MOTIVATION: Many tasks in sequence analysis ask to identify biologically related sequences in a large set. The edit distance, being a sensible model for both evolution and sequencing error, is widely used in these tasks as a measure. The resulting computational problem-to recognize all pairs of sequences within a small edit distance-turns out to be exceedingly difficult, since the edit distance is known to be notoriously expensive to compute and that all-versus-all comparison is simply not acceptable with millions or billions of sequences. Among many attempts, we recently proposed the locality-sensitive bucketing (LSB) functions to meet this challenge. Formally, a (d1,d2)-LSB function sends sequences into multiple buckets with the guarantee that pairs of sequences of edit distance at most d1 can be found within a same bucket while those of edit distance at least d2 do not share any. LSB functions generalize the locality-sensitive hashing (LSH) functions and admit favorable properties, with a notable highlight being that optimal LSB functions for certain (d1,d2) exist. LSB functions hold the potential of solving above problems optimally, but the existence of LSB functions for more general (d1,d2) remains unclear, let alone constructing them for practical use. RESULTS: In this work, we aim to utilize machine learning techniques to train LSB functions. With the development of a novel loss function and insights in the neural network structures that can potentially extend beyond this specific task, we obtained LSB functions that exhibit nearly perfect accuracy for certain (d1,d2), matching our theoretical results, and high accuracy for many others. Comparing to the state-of-the-art LSH method Order Min Hash, the trained LSB functions achieve a 2- to 5-fold improvement on the sensitivity of recognizing similar sequences. An experiment on analyzing erroneous cell barcode data is also included to demonstrate the application of the trained LSB functions. AVAILABILITY AND IMPLEMENTATION: The code for the training process and the structure of trained models are freely available at https://github.com/Shao-Group/lsb-learn.

MOTIVATION: Modern methods for computation-intensive tasks in sequence analysis (e.g. read mapping, sequence alignment, genome assembly, etc.) often first transform each sequence into a list of short, regular-length seeds so that compact data structures and efficient algorithms can be employed to handle the ever-growing large-scale data. Seeding methods using kmers (substrings of length k) have gained tremendous success in processing sequencing data with low mutation/error rates. However, they are much less effective for sequencing data with high error rates as kmers cannot tolerate errors. RESULTS: We propose SubseqHash, a strategy that uses subsequences, rather than substrings, as seeds. Formally, SubseqHash maps a string of length n to its smallest subsequence of length k, k < n, according to a given order overall length-k strings. Finding the smallest subsequence of a string by enumeration is impractical as the number of subsequences grows exponentially. To overcome this barrier, we propose a novel algorithmic framework that consists of a specifically designed order (termed ABC order) and an algorithm that computes the minimized subsequence under an ABC order in polynomial time. We first show that the ABC order exhibits the desired property and the probability of hash collision using the ABC order is close to the Jaccard index. We then show that SubseqHash overwhelmingly outperforms the substring-based seeding methods in producing high-quality seed-matches for three critical applications: read mapping, sequence alignment, and overlap detection. SubseqHash presents a major algorithmic breakthrough for tackling the high error rates and we expect it to be widely adapted for long-reads analysis. AVAILABILITY AND IMPLEMENTATION: SubseqHash is freely available at https://github.com/Shao-Group/subseqhash.

Many bioinformatics applications involve bucketing a set of sequences where each sequence is allowed to be assigned into multiple buckets. To achieve both high sensitivity and precision, bucketing methods are desired to assign similar sequences into the same bucket while assigning dissimilar sequences into distinct buckets. Existing $k$-mer-based bucketing methods have been efficient in processing sequencing data with low error rate, but encounter much reduced sensitivity on data with high error rate. Locality-sensitive hashing (LSH) schemes are able to mitigate this issue through tolerating the edits in similar sequences, but state-of-the-art methods still have large gaps. Here we generalize the LSH function by allowing it to hash one sequence into multiple buckets. Formally, a bucketing function, which maps a sequence (of fixed length) into a subset of buckets, is defined to be $(d_1, d_2)$-sensitive if any two sequences within an edit distance of $d_1$ are mapped into at least one shared bucket, and any two sequences with distance at least $d_2$ are mapped into disjoint subsets of buckets. We construct locality-sensitive bucketing (LSB) functions with a variety of values of $(d_1,d_2)$ and analyze their efficiency with respect to the total number of buckets needed as well as the number of buckets that a specific sequence is mapped to. We also prove lower bounds of these two parameters in different settings and show that some of our constructed LSB functions are optimal. These results provide theoretical foundations for their practical use in analyzing sequences with high error rate while also providing insights for the hardness of designing ungapped LSH functions.

Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between the total length of $P$ and the distance of its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For a parameter $c \geq 1$, we call $P$ a $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$ for every triple $(i,j,k)$, $1 \leq i 0$, there is a noncrossing $c$-chain that has stretch factor $\Omega(n^{1/2-\varepsilon})$ for sufficiently large constant $c=c(\varepsilon)$; (ii) on the other hand, the stretch factor of a $c$-chain $P$ is $O(n^{1/2})$ for every constant $c\geq 1$, regardless of whether $P$ is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain $P$ in $\mathbb{R}^2$ with $n$ vertices, the minimum $c\geq 1$ for which $P$ is a $c$-chain in $O(n^{2.5}\ {\rm polylog}\ n)$ expected time and $O(n\log n)$ space. These results generalize to $\mathbb{R}^d$. For every dimension $d\geq 2$ and every $\varepsilon>0$, we construct a noncrossing $c$-chain that has stretch factor $\Omega(n^{(1-\varepsilon)(d-1)/d})$; on the other hand, the stretch factor of any $c$-chain is $O((n-1)^{(d-1)/d})$; for every $c>1$, we can test whether an $n$-vertex chain in $\mathbb{R}^d$ is a $c$-chain in $O(n^{3-1/d}\ {\rm polylog}\ n)$ expected time and $O(n\log n)$ space.

Given a sequence $A$ of $n$ numbers and an integer (target) parameter $1\leq i\leq n$, the (exact) selection problem asks to find the $i$-th smallest element in $A$. An element is said to be $(i,j)$-mediocre if it is neither among the top $i$ nor among the bottom $j$ elements of $S$. The approximate selection problem asks to find a $(i,j)$-mediocre element for some given $i,j$; as such, this variant allows the algorithm to return any element in a prescribed range. In the first part, we revisit the selection problem in the two-party model introduced by Andrew Yao (1979) and then extend our study of exact selection to the multiparty model. In the second part, we deduce some communication complexity benefits that arise in approximate selection. In particular, we present a deterministic protocol for finding an approximate median among $k$ players.

Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space.