Maximum Matching Research Articles

Shared-Memory Parallel Edmonds Blossom Algorithm for Maximum Cardinality Matching in General Graphs.

The Edmonds Blossom algorithm is implemented here using depth-first search, which is intrinsically serial. By streamlining the code, our serial implementation is consistently three to five times faster than the previously fastest general graph matching code. By extracting parallelism across iterations of the algorithm, with coarse-grain locking, we are able to further reduce the run time on random regular graphs four-fold and obtain a two-fold reduction of run time on real-world graphs with similar topology. Solving very sparse graphs (average degree less than four) exhibiting community structure with eight threads led to a slow down of three-fold, but this slow down is replaced by marginal speed up once the average degree is greater than four. We conclude that our parallel coarse-grain locking implementation performs well when extracting parallelism from this augmenting-path-based algorithm and may work well for similar algorithms.

MEM-based pangenome indexing for k-mer queries.

Pangenomes are growing in number and size, thanks to the prevalence of high-quality long-read assemblies. However, current methods for studying sequence composition and conservation within pangenomes have limitations. Methods based on graph pangenomes require a computationally expensive multiple-alignment step, which can leave out some variation. Indexes based on k-mers and de Bruijn graphs are limited to answering questions at a specific substring length k. We present Maximal Exact Match Ordered (MEMO), a pangenome indexing method based on maximal exact matches (MEMs) between sequences. A single MEMO index can handle arbitrary-length queries over pangenomic windows. MEMO enables both queries that test k-mer presence/absence (membership queries) and that count the number of genomes containing k-mers in a window (conservation queries). MEMO's index for a pangenome of 89 human autosomal haplotypes fits in 2.04 GB, 8.8× smaller than a comparable KMC3 index and 11.4× smaller than a PanKmer index. MEMO indexes can be made smaller by sacrificing some counting resolution, with our decile-resolution HPRC index reaching 0.67 GB. MEMO can conduct a conservation query for 31-mers over the human leukocyte antigen locus in 13.89 seconds, 2.5x faster than other approaches. MEMO's small index size, lack of k-mer length dependence, and efficient queries make it a flexible tool for studying and visualizing substring conservation in pangenomes.

Computing balanced solutions for large international kidney exchange schemes

To overcome incompatibility issues, kidney patients may swap their donors. In international kidney exchange programmes (IKEPs), countries merge their national patient–donor pools. We consider a recently introduced credit system. In each round, countries are given an initial “fair” allocation of the total number of kidney transplants. This allocation is adjusted by a credit function yielding a target allocation. The goal is to find a solution that approaches the target allocation as closely as possible, to ensure long-term stability of the international pool. As solutions, we use maximum matchings that lexicographically minimize the country deviations from the target allocation. We perform, for the first time, a computational study for a large number of countries. For the initial allocations we use two easy-to-compute solution concepts, the benefit value and the contribution value, and four classical but hard-to-compute concepts, the Shapley value, nucleolus, Banzhaf value and tau value. By using state-of-the-art software we show that the latter four concepts are now within reach for IKEPs of up to fifteen countries. Our experiments show that using lexicographically minimal maximum matchings instead of ones that only minimize the largest deviation from the target allocation (as previously done) may make an IKEP up to 54% more balanced.

Multifunctional in vitro, in silico and DFT analyses on antimicrobial BagremycinA biosynthesized by Micromonospora chokoriensis CR3 from Hieracium canadense

Among the actinomycetes in the rare genera, Micromonospora is of great interest since it has been shown to produce novel therapeutic compounds. Particular emphasis is now on its isolation from plants since its population from soil has been extensively explored. The strain CR3 was isolated as an endophyte from the roots of Hieracium canadense, and it was identified as Micromonospora chokoriensis through 16S gene sequencing and phylogenetic analysis. The in-vitro analysis of its extract revealed it to be active against the clinical isolates of methicillin-resistant Staphylococcus aureus (MRSA) and Candida tropicalis (15 mm). No bioactivity was observed against Gram-negative bacteria, Escherichia coli ATCC 25922, and Klebsiella pneumoniae ATCC 706003. The Micromonospora chokoriensis CR3 extract was also analyzed through the HPLC-DAD-UV–VIS resident database, and it gave a maximum match factor of 997.334 with the specialized metabolite BagremycinA (BagA). The in-silico analysis indicated that BagA strongly interacted with the active site residues of the sterol 14-α demethylase and thymidylate kinase enzymes, with the lowest binding energies of − 9.7 and − 8.3 kcal/mol, respectively. Furthermore, the normal mode analysis indicated that the interaction between these proteins and BagA was stable. The DFT quantum chemical properties depicted BagA to be reasonably reactive with a HOMO-LUMO gap of (ΔE) of 4.390 eV. BagA also passed the drug-likeness test with a synthetic accessibility score of 2.06, whereas Protox-II classified it as a class V toxicity compound with high LD50 of 2644 mg/kg. The current study reports an endophytic actinomycete, M. chokoriensis, associated with H. canadense producing the bioactive metabolite BagA with promising antimicrobial activity, which can be further modified and developed into a safe antimicrobial drug.

A Theory of Alternating Paths and Blossoms from the Perspective of Minimum Length

The Micali–Vazirani (MV) algorithm for finding a maximum cardinality matching in general graphs, which was published in 1980, remains to this day the most efficient known algorithm for the problem. The current paper gives the first complete and correct proof of this algorithm. The MV algorithm resorts to finding minimum-length augmenting paths. However, such paths fail to satisfy an elementary property, called breadth first search honesty in this paper. In the absence of this property, an exponential time algorithm appears to be called for—just for finding one such path. On the other hand, the MV algorithm accomplishes this and additional tasks in linear time. The saving grace is the various “footholds” offered by the underlying structure, which the algorithm uses in order to perform its key tasks efficiently. The theory expounded in this paper elucidates this rich structure and yields a proof of correctness of the algorithm. It may also be of independent interest as a set of well-knit graph-theoretic facts. Funding: This work was supported in part by the National Science Foundation [Grant CCF-2230414].

Statistical methods to control for confounders in rare disease settings that use external control

ABSTRACT In the drug development for rare disease, the number of treated subjects in the clinical trial is often very small, whereas the number of external controls can be relatively large. There is no clear guidance on choosing an appropriate statistical method to control baseline confounding in this situation. To fill this gap, we conduct extensive simulations to evaluate the performance of commonly used matching and weighting methods as well as the more recently developed targeted maximum likelihood estimation (TMLE) and cardinality matching in small sample settings, mimicking the motivating data from a pediatric rare disease. Among the methods examined, the performance of coarsened exact matching (CEM) and TMLE are relatively robust under various model specifications. CEM is only feasible when the number of controls far exceeds the number of treated, whereas TMLE has better performance with less extreme treatment allocation ratios. Our simulations suggest bootstrap is useful for variance estimation in small samples after matching.

Accelerating spliced alignment of long RNA sequencing reads using parallel maximal exact match retrieval

The genomics landscape has undergone a revolutionary transformation with the emergence of third-generation sequencing technologies. Fueled by the exponential surge in sequencing data, there is an urgent demand for accurate and rapid algorithms to effectively handle this burgeoning influx. Under such circumstances, we developed a parallelized, yet accuracy-lossless algorithm for maximal exact match (MEM) retrieval to strategically address the computational bottleneck of uLTRA, a leading spliced alignment algorithm known for its precision in handling long RNA sequencing (RNA-seq) reads. The design of the algorithm incorporates a multi-threaded strategy, enabling the concurrent processing of multiple reads simultaneously. Additionally, we implemented the serialization of index required for MEM retrieval to facilitate its reuse, resulting in accelerated startup for practical tasks. Extensive experiments demonstrate that our parallel algorithm achieves significant improvements in runtime, speedup, throughput, and memory usage. When applied to the largest human dataset, the algorithm achieves an impressive speedup of 10.78 × , significantly improving throughput on a large scale. Moreover, the integration of the parallel MEM retrieval algorithm into the uLTRA pipeline introduces a dual-layered parallel capability, consistently yielding a speedup of 4.99 × compared to the multi-process and single-threaded execution of uLTRA. The thorough analysis of experimental results underscores the adept utilization of parallel processing capabilities and its advantageous performance in handling large datasets. This study provides a showcase of parallelized strategies for MEM retrieval within the context of spliced alignment algorithm, effectively facilitating the process of RNA-seq data analysis. The code is available at https://github.com/RongxingWong/AcceleratingSplicedAlignment.

Super-wideband fractal antenna of two-arm structure with broadband balun

ABSTRACT A balanced two-arm antenna has been proposed for super-wideband (SWB) operation to work efficiently over the frequency range (2.95 − 34 GHz ). The antenna proposed for SWB operation can be considered as composed of two main parts. The first part is the radiating surface of the antenna and is composed of two-arms leading to a balanced antenna structure. For wideband operation, each arm has a double-fractal (angular/radial) structure. The second part of the antenna structure is the wideband feeding balun that is responsible for realising maximum power transfer and impedance matching while feeding the balanced radiating two-arm structure from the conventional unbalanced coaxial line. A prototype has been fabricated for experimental characterisation of the proposed antenna. Both the simulation results and the experimental measurements come in good agreement with each other showing that the proposed antenna can be good candidate for many super-wideband applications of future generations of mobile communications. The proposed antenna has overall dimensions of 40.0 mm × 21.0 mm × 1.5 mm . It is shown by simulation and measurement that the proposed antenna has %BW of 168 % , RBW of 11.5 : 1 , and BDR of 2015 . The maximum gain ranges between 2.3 and 6.2 dBi. The radiation efficiency is greater than 98% over the frequency range ( 2 − 24 GHz ) and is greater than 85% over the frequency range ( 24 − 34 GHz ).

Fractional Domination Number of the Fractal Graphs

Background/Objectives: A fractal is a collection of clearly defined structures that are uneven or irregular and can be divided into smaller pieces. Each of these smaller pieces has a relationship to the overall structure at random ranges that are analogous or identical. Fractal graphs are a highly effective way to construct well-defined patterns. The fractional domination number in a graph refers to the weight of a minimum fractional dominating function. The study is to find the bounds of the fractional domination and their related parameters in the fractal graphs. Methods: The sharp bounds and the constant ratio of fractional domination and their related parameters were calculated using an iterative process in all iterations of a self-similarity fractal graph. Findings: The study established the relation between the fractional domination number and other relevant parameters as fractional independence domination numbers for each iteration in the fractal graphs and explained how the concept of fractional domination works on fractal graphs such as the Koch snowflake and the Sierpinski triangle. Novelty: Previous researchers have studied bounds on maximum matching and dominating sets in fractal graphs such as the Koch snowflake and the Sierpinski triangle. The present study enhances the idea of finding the bounds of fractional domination parameters in a fractal graph. The construction of new beautiful ornaments, locating damaged cells for drug injection in medical science, placing stones in jewels, plumping work, and designing fractal antennas all can be made possible by this work in self-similarity fractal graphs. Keywords: Fractals, Fractional Dominating Functions, Fractional Domination Number, Fractional Independence Domination Number, Fractional Domination Chain

Domination parameters and the removal of matchings

We consider the effect of the removal of a matching from a graph on the value of domination and domination-related parameters with emphasis on the removal of a maximum matching from a cubic graph or tree. For example, we show that the independent domination number of a nontrivial tree increases on the removal of any maximum matching.

The average size of maximal matchings in graphs

The average size of maximal matchings in graphs

Strong core and Pareto-optimality in the multiple partners matching problem under lexicographic preference domains

We study strong core and Pareto-optimal solutions for multiple partners matching problem under lexicographic preference domains from a computational point of view. The restriction to the two-sided case is called stable many-to-many matching problem and the general one-sided case is called stable fixtures problem. We provide an example to show that the strong core can be empty even for many-to-many problems, and that deciding the non-emptiness of the strong core is NP-hard. On the positive side, we give efficient algorithms for finding a near feasible strong core solution and for finding a fractional matching in the strong core of fractional matchings. In contrast with the NP-hardness result for the stable fixtures problem, we show that finding a maximum size matching that is Pareto-optimal can be done efficiently for many-to-many problems. Finally, we show that for reverse-lexicographic preferences the strong core is always non-empty in the many-to-many case.

Parameterization of (Partial) Maximum Satisfiability above Matching in a Variable-Clause Graph

In the paper, we study the Maximum Satisfiability and the Partial Maximum Satisfiability problems. Using Gallai–Edmonds decomposition, we significantly improve the upper bound for the Maximum Satisfiability problem parameterized above maximum matching in the variable-clause graph. Our algorithm operates with a runtime of O*(2.83^k'), a substantial improvement compared to the previous approach requiring O*(4^k' ), where k' denotes the relevant parameter. Moreover, this result immediately implies O*(1.14977^m) and O*(1.27895^m) time algorithms for the (n, 3)-MaxSAT and (n, 4)-MaxSAT where m is the overall number of clauses. These upper bounds improve prior-known upper bounds equal to O*(1.1554^m) and O*(1.2872^m). We also adapt the algorithm so that it can handle instances of Partial Maximum Satisfiability without losing performance in some cases. Note that this is somewhat surprising, as the existence of even one hard clause can significantly increase the hardness of a problem.

An efficient two-heuristic algorithm for the student-project allocation with preferences over projects

The Student-Project Allocation with preferences over Projects problem is a many-to-one stable matching problem that aims to assign students to projects in project-based courses so that students and lecturers meet their preference and capacity constraints. In this paper, we propose an efficient two-heuristic algorithm to solve this problem. Our algorithm starts from an empty matching and iteratively constructs a maximum stable matching of students to projects. At each iteration, our algorithm finds an unassigned student and assigns her/his most preferred project to her/him to form a student-project pair in the matching. If the project or the lecturer who offered the project is over-subscribed, our algorithm uses two heuristic functions, one for the over-subscribed project and the other for the over-subscribed lecturer, to remove a student-project pair in the matching. To reach a stable matching of a maximum size, our two heuristics are designed such that the removed student has the most opportunities to be assigned to some project in the next iterations. Experimental results show that our algorithm is efficient in execution time and solution quality for solving the problem.

Faster Maximal Exact Matches with Lazy LCP Evaluation.

MONI (Rossi et al., JCB 2022) is a BWT-based compressed index for computing the matching statistics and maximal exact matches (MEMs) of a pattern (usually a DNA read) with respect to a highly repetitive text (usually a database of genomes) using two operations: LF-steps and longest common extension (LCE) queries on a grammar-compressed representation of the text. In practice, most of the operations are constant-time LF-steps but most of the time is spent evaluating LCE queries. In this paper we show how (a variant of) the latter can be evaluated lazily, so as to bound the total time MONI needs to process the pattern in terms of the number of MEMs between the pattern and the text, while maintaining logarithmic latency.

Optimal Matching with Matching Priority

Matching algorithms are commonly used to build comparable subsets (matchings) in observational studies. When a complete matching is not possible, some units must necessarily be excluded from the final matching. This may bias the final estimates comparing the two populations, and thus it is important to reduce the number of drops to avoid unsatisfactory results. Greedy matching algorithms may not reach the maximum matching size, thus dropping more units than necessary. Optimal matching algorithms do ensure a maximum matching size, but they implicitly assume that all units have the same matching priority. In this paper, we propose a matching strategy which is order optimal in the sense that it finds a maximum matching size which is consistent with a given matching priority. The strategy is based on an order-optimal matching algorithm originally proposed in connection with assignment problems by D. Gale. When a matching priority is given, the algorithm ensures that the discarded units have the lowest possible matching priority. We discuss the algorithm’s complexity and its relation with classic optimal matching. We illustrate its use with a problem in a case study concerning a comparison of female and male executives and a simulation.

Diverse Pairs of Matchings

AbstractWe initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph G and an integer k, ask whether G has two (maximum/perfect) matchings whose symmetric difference is at least k. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is $$\textsf{NP}$$ NP -complete on general graphs if k is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is $$\textsf{FPT}$$ FPT parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on $${\mathcal {O}}(k^2)$$ O ( k 2 ) vertices.

Finding maximal exact matches in graphs

BackgroundWe study the problem of finding maximal exact matches (MEMs) between a query string Q and a labeled graph G. MEMs are an important class of seeds, often used in seed-chain-extend type of practical alignment methods because of their strong connections to classical metrics. A principled way to speed up chaining is to limit the number of MEMs by considering only MEMs of length at least κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} (κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}-MEMs). However, on arbitrary input graphs, the problem of finding MEMs cannot be solved in truly sub-quadratic time under SETH (Equi et al., TALG 2023) even on acyclic graphs.ResultsIn this paper we show an O(n·L·dL-1+m+Mκ,L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\cdot L \\cdot d^{L-1} + m + M_{\\kappa ,L})$$\\end{document}-time algorithm finding all κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}-MEMs between Q and G spanning exactly L nodes in G, where n is the total length of node labels, d is the maximum degree of a node in G, m=|Q|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m = |Q|$$\\end{document}, and Mκ,L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{\\kappa ,L}$$\\end{document} is the number of output MEMs. We use this algorithm to develop a κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}-MEM finding solution on indexable Elastic Founder Graphs (Equi et al., Algorithmica 2022) running in time O(nH2+m+Mκ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(nH^2 + m + M_\\kappa )$$\\end{document}, where H is the maximum number of nodes in a block, and Mκ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_\\kappa$$\\end{document} is the total number of κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}-MEMs. Our results generalize to the analysis of multiple query strings (MEMs between G and any of the strings). Additionally, we provide some experimental results showing that the number of graph MEMs is an order of magnitude smaller than the number of string MEMs of the corresponding concatenated collection.ConclusionsWe show that seed-chain-extend type of alignment methods can be implemented on top of indexable Elastic Founder Graphs by providing an efficient way to produce the seeds between a set of queries and the graph. The code is available in https://github.com/algbio/efg-mems.

Applying RFID and NLP for efficient warehouse picking

This paper proposes an intelligent warehouse-picking approach using radio frequency identification (RFID) indoor positioning and natural language processing (NLP) speech recognition. A forward maximum matching algorithm segments speech into domain terminology. Location was estimated by RFID signal strengths between reference tags and pickers. Simulation results demonstrated a 50% reduction in segmentation runtime versus conventional methods. Speech recognition accuracy reached 90–95%, improving by 23% over baseline. Positioning accuracy also increased substantially. The techniques can reduce picking errors and costs. Further work should evaluate performance in real-world environments.

Sharp lower bounds for the number of maximum matchings in bipartite multigraphs

AbstractWe study the minimum number of maximum matchings in a bipartite multigraph with parts and under various conditions, refining the well‐known lower bound due to M. Hall. When , every vertex in has degree at least , and every vertex in has at least distinct neighbors, the minimum is when and is when . When every vertex has at least two neighbors and , the minimum is , where . We also determine the minimum number of maximum matchings in several other situations. We provide a variety of sharpness constructions.