Maximum Matching Research Articles

New Results on Graph Matching from Degree-Preserving Growth

The recently introduced model in S. R. Kharel et al.’s study [Degree-preserving network growth. Nature Physics 2022, 18, 100–106] uses matchings to insert new vertices of prescribed degrees into the current graph of an ever-growing graph sequence. The process depends both on the size of the largest available matching, which is the focus of this paper, as well as on the actual choice of the matching. Here, we first show that the question of whether a graphic degree sequence, extended with a new degree 2δ, remains graphic and is equivalent to the existence of a realization of the original degree sequence with a matching of size δ. Secondly, we present lower bounds for the size of the maximum matchings in any realization of the degree sequence. We then study the bounds on the size of maximal matchings in some realizations of the sequence, known as the potential matching number. We also estimate the minimum size of both maximal and maximum matchings, as determined by the degree sequence, independently of graphical realizations. Along this line we answer a question raised by T. Biedl et al.: Tight bounds on maximal and maximum matchings. Discrete Mathematics 2004, 285, 7–15.

Arabic word tokenization system using the maximum matching model

Word tokenization is the first stage for higher-order Natural Language Processing (NLP) tasks like Part-of-Speech (PoS) tagging, parsing, and named entity recognition. The amount of text on the World Wide Web is growing daily in the present era of technology, necessitating the use of advanced instruments. Since more and more people speak Arabic around the world, Arabic language processing systems must be improved. Due to the writing style of Arabic with a lack of support for capitalization features and the use of compound words, it is difficult to perform word tokenization. This research paper proposes a novel Arabic word tokenization system based on the knowledge. To develop this system, a maximum matching model with its two variations, namely forward and reverse maximum matching is used. The proposed system is implemented in Python. The results produced during system evaluation report high performance.

Multimodal Data Mining Based on Self Attention Feature Alignment

This study explores the application of multimodal data mining in text and image vector processing, aiming to improve the depth and breadth of data analysis by integrating information from different data types. We use the FairFace dataset combined with the CLIP model encoding layer to obtain text and image vectors, and use the K-Means clustering algorithm to achieve vector dimensionality reduction. Subsequently, we introduced the bipartite graph matching algorithm to achieve maximum matching between text vectors and image vectors, and calculated the contrastive learning loss and similarity loss. The entire process covers steps such as data preparation, feature extraction, vector dimensionality reduction, matching algorithms, and loss assessment, constructing a complete text and image matching task process. Our research contributions include using K-Means clustering algorithm to achieve vector dimensionality reduction, as well as introducing bipartite graph matching algorithm and calculating two types of losses in text and image vector matching, further improving matching quality.

Computing MEMs and Relatives on Repetitive Text Collections

We consider the problem of computing the Maximal Exact Matches (MEMs) of a given pattern \(P[1\mathinner{.. }m]\) on a large repetitive text collection \(T[1\mathinner{.. }n]\) over an alphabet of size \(\sigma\) , which is represented as a (hopefully much smaller) run-length context-free grammar of size \(g_{rl}\) . We show that the problem can be solved in time \(O(m^{2}\log^{\epsilon}n)\) , for any constant \(\epsilon\gt0\) , on a data structure of size \(O(g_{rl})\) . Further, on a locally consistent grammar of size \(O(\delta\log\frac{n\log\sigma}{\delta\log n})\) , the time decreases to \(O(m\log m(\log m+\log^{\epsilon}n))\) . The value \(\delta\) is a function of the substring complexity of \(T\) and \(\Omega(\delta\log\frac{n\log\sigma}{\delta\log n})\) is a tight lower bound on the compressibility of repetitive texts \(T\) , so our structure has optimal size in terms of \(n\) , \(\sigma\) , and \(\delta\) . We extend our results to several related problems, such as finding \(k\) -MEMs, MUMs, rare MEMs, and applications. Categories and Subject Descriptors: E.1 [Data structures] ; E.2 [Data storage representations] ; E.4 [Coding and information theory]: Data compaction and compression; F.2.2 [Analysis of algorithms and problem complexity] : Nonnumerical algorithms and problems— Pattern matching, Computations on discrete structures, Sorting and searching

Euclidean Maximum Matchings in the Plane—Local to Global

Euclidean Maximum Matchings in the Plane—Local to Global

Modelling of insulating potential in ultra-thin (42 å) silicon oxide film

Based on previously conducted measurements of the tunneling current-voltage characteristics of metal-SiO2-Si (MOS) structures, modeling of the insulating potential in an ultra-thin (4.2 nm) silicon oxide film was performed. The potential in the dielectric was defined in the shape of a trapezoid, with the lateral slopes simulating transition layers and the top base representing the bulk of SiO2. The model parameters – the barrier height and the coordinates of the trapezoid's corner points – were calculated to achieve the maximum match between the experimental and theoretical voltage derivatives of the current logarithm. Common features of the insulating potential, similar to those in thinner silicon oxide films (3.7 nm), were identified: the barrier occupies up to half of the nominal volume of the dielectric gap and is shifted towards the gate electrode, with its slope towards the semiconductor substrate being much more gradual compared to the slope adjacent to the gate.

Basilica: New canonical decomposition in matching theory

AbstractIn matching theory, one of the most fundamental and classical branches of combinatorics, canonical decompositions of graphs are powerful and versatile tools that form the basis of this theory. However, the abilities of the known canonical decompositions, that is, the Dulmage–Mendelsohn, Kotzig–Lovász, and Gallai–Edmonds decompositions, are limited because they are only applicable to particular classes of graphs, such as bipartite graphs, or they are too sparse to provide sufficient information. To overcome these limitations, we introduce a new canonical decomposition that is applicable to all graphs and provides much finer information. This decomposition also provides the answer to the longstanding absence of a canonical decomposition that is nontrivially applicable to general graphs with perfect matchings. We focus on the notion of factor‐components as the fundamental building blocks of a graph; through the factor‐components, our new canonical decomposition states how a graph is organized and how it contains all the maximum matchings. The main results that constitute our new theory are the following: (i) a canonical partial order over the set of factor‐components, which describes how a graph is constructed from its factor‐components; (ii) a generalization of the Kotzig–Lovász decomposition, which shows the inner structure of each factor‐component in the context of the entire graph; and (iii) a canonically described interrelationship between (i) and (ii), which integrates these two results into a unified theory of a canonical decomposition. These results are obtained in a self‐contained way, and our proof of the generalized Kotzig–Lovász decomposition contains a shortened and self‐contained proof of the classical counterpart.

Dynamic Matching with Better-than-2 Approximation in Polylogarithmic Update Time

We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than 2 . Specifically, we obtain a \(1+\tfrac{1}{\sqrt {2}}+\epsilon \approx 1.707+\epsilon\) approximation in bipartite graphs and a \(1.973+\epsilon\) approximation in general graphs. We thus answer in the affirmative the value version of the major open question repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms’ approximation and worst-case update time bounds both hold w.h.p. against adaptive adversaries. Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS’21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.

Efficient incremental training using a novel NMT-SMT hybrid framework for translation of low-resource languages.

The data-hungry statistical machine translation (SMT) and neural machine translation (NMT) models offer state-of-the-art results for languages with abundant data resources. However, extensive research is imperative to make these models perform equally well for low-resource languages. This paper proposes a novel approach to integrate the best features of the NMT and SMT systems for improved translation performance of low-resource English-Tamil language pair. The suboptimal NMT model trained with the small parallel corpus translates the monolingual corpus and selects only the best translations, to retrain itself in the next iteration. The proposed method employs the SMT phrase-pair table to determine the best translations, based on the maximum match between the words of the phrase-pair dictionary and each of the individual translations. This repeating cycle of translation and retraining generates a large quasi-parallel corpus, thus making the NMT model more powerful. SMT-integrated incremental training demonstrates a substantial difference in translation performance as compared to the existing approaches for incremental training. The model is strengthened further by adopting a beam search decoding strategy to produce k best possible translations for each input sentence. Empirical findings prove that the proposed model with BLEU scores of 19.56 and 23.49 outperforms the baseline NMT with scores 11.06 and 17.06 for Eng-to-Tam and Tam-to-Eng translations, respectively. METEOR score evaluation further corroborates these results, proving the supremacy of the proposed model.

Maximum Cardinality \(f\) -Matching in Time \(O(n^{2/3}m)\)

We present an algorithm that finds a maximum cardinality \(f\) -matching of a simple graph in time \(O(n^{2/3}m)\) . Here \(f:V\to\mathbb{N}\) is a given function and an \(f\) -matching is a subgraph wherein each vertex \(v\in V\) has degree \(\leq f(v)\) . This result generalizes a string of algorithms that concentrate on simple bipartite graphs. The bipartite case is based on the notion of level graph, introduced by Dinic for network flow. In general graphs this notion breaks down: Vertices no longer have unique levels, and there are too many levels to analyze the corresponding level graph like bipartite graphs ( \(\Theta(n^{2})\) vs. \(n\) ). We use “natural” levels to prove properties of shortest augmenting trails (e.g., formulas for trail length). We use “shortened” levels to derive the algorithm's time bound. The algorithm, unmodified, is also efficient on multigraphs, achieving time \(O(\min\{\sqrt{f(V)},n\} m)\) for \(f(V)=\sum_{v}f(v)\) . The special case \(f\equiv 1\) shows the algorithm duplicates the classic time bound for maximum cardinality matching, \(O(\sqrt{n} m)\) .

A D2D user pairing algorithm based on motion prediction and power control

AbstractUser pairing plays an important role in device‐to‐device (D2D) relay communication, contributing significantly to maintaining low energy consumption, high throughput, and overall energy efficiency in the communication system. To achieve these purposes, an attention‐based long short‐term memory motion prediction model (AT‐LSTM) and propose a joint power control algorithm. Leveraging these techniques, we also propose a D2D user pairing algorithm, distance–power–SINR pairing algorithm (DPSPA), which comprehensively considers factors such as D2D communication distances, transmit power, and signal‐to‐interference‐plus‐noise ratio. Initially, the AT‐LSTM model is utilized to predict the location of users. Subsequently, the distance between the user terminal device and each communication point and the base station, filtering cache points, and non‐cache points within the D2D communication radius are calculated. Then, based on the distance, required transmission power, and signal‐to‐interference‐plus‐noise ratio of each point, the evaluation index (the best matching product) is obtained. Finally, the point with the maximum best matching product is selected for D2D direct communication mode, D2D relay communication mode, or cellular communication mode. Simulation results demonstrate that DPSPA effectively reduces system energy consumption, enhances system throughput, and improves overall energy efficiency.

Maximum radial pattern matching for minimum star map identification

This paper proposes an all-sky star map identification algorithm that can simultaneously achieve high identification probability, low algorithm complexity, and small databases for well photometric and intrinsic parameters-calibrated star sensors. The proposed algorithm includes three main steps. First, a binary radial pattern table is constructed offline. Then, the maximum value matching of the radial pattern is performed between the star spots and the guide stars, and the star pairs (i.e., the minimum star map) after radial pattern matching undergo a coarse matching through angular distance cross-validation. Finally, a reference star map is designed based on the identified star pairs, and the matching of all the star spots in the field of view is realized. Simulation and analysis results show that the database required by the proposed algorithm for 5,000 guide stars is not larger than 200 KB. Also, when false and missing star spots account for 50% of all guide stars and the star spot extraction error is 0.5 pixel (the corresponding pointing error is 26″), the average star map identification time of the proposed algorithm is less than 2 ms, and its identification probability is higher than 98%. The results demonstrate that the proposed algorithm performs better than similar algorithms.

Dichotomies for Maximum Matching Cut: H-freeness, bounded diameter, bounded radius

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs. A disconnected perfect matching of a graph G is a perfect matching that contains a matching cut of G. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for (P6+sP2)-free graphs for every s≥0, extending a known result for P5-free graphs (Bouquet and Picouleau, 2020).

Quick evaluation and regulation of the maximum instantaneous power and matching resistance for droplet-based electricity generators

Quick evaluation and regulation of the maximum instantaneous power and matching resistance for droplet-based electricity generators

How to Find Long Maximal Exact Matches and Ignore Short Ones.

Finding maximal exact matches (MEMs) between strings is an important task in bioinformatics, but it is becoming increasingly challenging as geneticists switch to pangenomic references. Fortunately, we are usually interested only in the relatively few MEMs that are longer than we would expect by chance. In this paper we show that under reasonable assumptions we can find all MEMs of length at least between a pattern of length and a text of length in time plus extra time only for each MEM of length at least nearly using a compact index for the text, suitable for pangenomics.

On the complexity of minimum maximal acyclic matchings

Low-Acy-Matching asks to find a maximal matching M in a given graph G of minimum cardinality such that the set of M-saturated vertices induces an acyclic subgraph in G. The decision version of Low-Acy-Matching is known to be NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{NP}}$$\\end{document}-complete. In this paper, we strengthen this result by proving that the decision version of Low-Acy-Matching remains NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{NP}}$$\\end{document}-complete for bipartite graphs with maximum degree 6 and planar perfect elimination bipartite graphs. We also show the hardness difference between Low-Acy-Matching and Max-Acy-Matching. Furthermore, we prove that, even for bipartite graphs, Low-Acy-Matching cannot be approximated within a ratio of n1-ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{1-\\epsilon }$$\\end{document} for any ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document} unless P=NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{P}}={\ extsf{NP}}$$\\end{document}. Finally, we establish that Low-Acy-Matching exhibits APX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{APX}$$\\end{document}-hardness when restricted to 4-regular graphs.

Taxonomic classification with maximal exact matches in KATKA kernels and minimizer digests.

For taxonomic classification, we are asked to index the genomes in a phylogenetic tree such that later, given a DNA read, we can quickly choose a small subtree likely to contain the genome from which that read was drawn. Although popular classifiers such as Kraken use -mers, recent research indicates that using maximal exact matches (MEMs) can lead to better classifications. For example, we can ■ build an augmented FM-index over the the genomes in the tree concatenated in left-to-right order; ■ for each MEM in a read, find the interval in the suffix array containing the starting positions of that MEM's occurrences in those genomes; ■ find the minimum and maximum values stored in that interval; ■ take the lowest common ancestor (LCA) of the genomes containing the characters at those positions. This solution is practical, however, only when the total size of the genomes in the tree is fairly small. In this paper we consider applying the same solution to three lossily compressed representations of the genomes' concatenation: ■ a KATKA kernel, which discards characters that are not in the first or last occurrence of any -tuple, for a parameter ; a minimizer digest; ■ a KATKA kernel of a minimizer digest. With a test dataset and these three representations of it, simulated reads and various parameter settings, we checked how many reads' longest MEMs occurred only in the sequences from which those reads were generated ("true positive" reads). For some parameter settings we achieved significant compression while only slightly decreasing the true-positive rate.

Almost tight bounds for online hypergraph matching

In the online hypergraph matching problem, hyperedges of size k over a common ground set arrive online in adversarial order. The goal is to obtain a maximum matching (disjoint set of hyperedges). A naïve greedy algorithm for this problem achieves a competitive ratio of 1k. We show that no (randomized) online algorithm has competitive ratio better than 2+o(1)k. If edges are allowed to be assigned fractionally, we give a deterministic online algorithm with competitive ratio 1−o(1)ln⁡(k) and show that no online algorithm can have competitive ratio strictly better than 1+o(1)ln⁡(k). Lastly, we give a 1−o(1)ln⁡(k) competitive algorithm for the fractional edge-weighted version of the problem under a free disposal assumption.

Clustering Integrated Fusion Model Based on the Maximum Matching Problem

The maximum-weight perfect matching inverse issue in graph theory and text clustering are the two primary topics of this study. We suggest a novel approach to text clustering that makes use of self-encoders and BERT embeddings for feature extraction in order to increase clustering accuracy. According to experimental results, our technique enhances the clustering results greatly and performs well on numerous short text datasets. In the context of graph theory, we examine the unit paradigm inverse issue of maximum-weight perfect matching with value constraints and provide a robust polynomial-time method for its solution. In addition to effectively solving the maximum-weight perfect matching inverse issue, our technique can also produce the best weight vector configuration scheme for real-world uses. In conclusion, our work has advanced the domains of text clustering and graph theory significantly, offering fresh approaches and theoretical underpinnings for future investigations.

Design and evaluation of a symmetric amplification mechanism based anthropomorphic shoulder

Shoulder joints determine the motion range of the upper limb. Thus, the compact and well-stiffened spherical parallel mechanism (SPM) has emerged as the mainstream shoulder prosthesis design approaches. However, the SPM’s moving pairs of redundant motions impose excessive constraints that limit its workspace. Therefore, amplifying the workspace of the SPM to cover the motion range required by human daily activities is a pressing problem in shoulder prosthesis design. To address this challenge, this study proposed a workspace amplification approach through the kinematic analysis of a symmetrically arranged 2 degrees of freedom (DoFs) passive mechanism, together with the designed and optimized 3-RRR SPM, to construct an anthropomorphic shoulder. The effectiveness and reliability of the proposed mechanism was verified through thorough analyses. Simulation and experiment results demonstrated that the workspace amplification mechanism could significantly increase the maximum motion match index between the shoulder prosthesis and the daily workspace of the human shoulder from only 26.3% to 94.79%, solving the problem that the traditional SPM-based prostheses cannot satisfy the motion range required by daily activities. Moreover, the proposed mechanism has the potential to amplify the workspace of most parallel mechanisms with multiple DoFs after proper modification.