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Algorithmica ◽  
2022 ◽  
Author(s):  
Yaqiao Li ◽  
Vishnu V. Narayan ◽  
Denis Pankratov

2021 ◽  
Vol 23 (2) ◽  
pp. 13-22
Author(s):  
Debmalya Mandal ◽  
Sourav Medya ◽  
Brian Uzzi ◽  
Charu Aggarwal

Graph Neural Networks (GNNs), a generalization of deep neural networks on graph data have been widely used in various domains, ranging from drug discovery to recommender systems. However, GNNs on such applications are limited when there are few available samples. Meta-learning has been an important framework to address the lack of samples in machine learning, and in recent years, researchers have started to apply meta-learning to GNNs. In this work, we provide a comprehensive survey of different metalearning approaches involving GNNs on various graph problems showing the power of using these two approaches together. We categorize the literature based on proposed architectures, shared representations, and applications. Finally, we discuss several exciting future research directions and open problems.


2021 ◽  
pp. 423
Author(s):  
Mochamad Abdul Basir ◽  
Mohamad Aminudin ◽  
Nila Ubaidah ◽  
Imam Kusmaryono

GeoGebra provides many facilities that can help teachers in geometry animation and manipulation so that it can be used as a means of visualization and construction in finding mathematical concepts. Utilization of the GeoGebra program can provide students with a visual experience in exploring graphs of quadratic functions in mathematics learning. The use of GeoGebra software is not easy, for that we need assistance and guidance activities from experts that are packaged in the form of community service activities. Mentoring activities are carried out online for students of SMA Negeri 1 Pegandon, Kendal, Central Java, through the Zoom Meeting application. The first session is an introduction to GeoGebra and the slider tool in the GeoGebra application. The second session is an exploration of the graph characteristics of a quadratic function by manipulating GeoGebra. The results of the self-regulated learning response questionnaire show that 75% of students have the initiative to learn, 71.55% of students can diagnose learning needs, 75.86% of students can set learning targets, 71,55% of students view difficulties as challenges, 69.83% of students utilize relevant reference sources, 73.71% of students can set learning strategies, 74.14% of students can evaluate learning processes and outcomes, and 78.02% of students have the good self-concept. Through GeoGebra mentoring service activities, students can be motivated to solve quadratic function graph problems and increase learning independence.GeoGebra banyak memberikan fasilitas yang dapat membantu guru dalam animasi geometri dan manipulasi sehingga dapat dijadikan sebagai sarana visualisasi dan mengkonstruksi dalam menemukan konsep matematis. Pemanfaatan program GeoGebra dapat memberikan pengalaman kepada siswa dalam mengeksplorasi grafik fungsi kuadrat pada pembelajaran matematika. Penggunaan software GeoGebra yang tidak mudah sehingga membutuhkan kegiatan pendampingan dan bimbingan dari ahli. Kegiatan pendampingan dilakukan secara online terhadap siswa SMA Negeri 1 Pegandon kabupaten Kendal Jawa Tengah melalui aplikasi zoom meeting. Sesi pertama kegiatan pengabdian adalah pengenalan GeoGebra dan tool slider pada aplikasi GeoGebra dan pada sesi kedua berupa eksplorasi karakteristik grafik fungsi kuadrat dengan memanipulasi GeoGebra. Hasil angket kemandirian belajar menunjukkan bahwa 75% siswa mempunyai inisiatif belajar, 71,55% siswa dapat mendiagnosa kebutuhan belajar, 75,86% siswa dapat menetapkan target belajar, 71,55% siswa beranggapan kesulitan bagian dari tantangan, 69,83% siswa memilih sumber referensi, 73,71% siswa dapat menetapkan strategi belajar, 74,14% siswa dapat melakukan evaluasi hasil belajar, dan 78,02% siswa memiliki self-concept yang baik. Melalui program pengabdian pendampingan geogebra dapat menjadikan siswa termotivasi untuk menyelesaikan masalah grafik fungsi kuadrat dan meningkatkan kemandirian belajar


2021 ◽  
Author(s):  
Bertrand Marchand ◽  
Yann Ponty ◽  
Laurent Bulteau

Abstract Hard graph problems are ubiquitous in Bioinformatics, inspiring the design of specialized Fixed-Parameter Tractable algorithms, many of which rely on a combination of tree-decomposition and dynamic programming. The time/space complexities of such approaches hinge critically on low values for the treewidth tw of the input graph. In order to extend their scope of applicability, we introduce the Tree-Diet problem, i.e. the removal of a minimal set of edges such that a given tree-decomposition can be slimmed down to a prescribed treewidth tw. Our rationale is that the time gained thanks to a smaller treewidth in a parameterized algorithm compensates the extra post-processing needed to take deleted edges into account. Our core result is an FPT dynamic programming algorithm for Tree-Diet, using 2^O(tw)n time and space. We complement this result with parameterized complexity lower-bounds for stronger variants (e.g., NP-hardness when tw or tw − tw is constant). We propose a prototype implementation for our approach which we apply on difficult instances of selected RNA-based problems: RNA design, sequence-structure alignment, and search of pseudoknotted RNAs in genomes, revealing very encouraging results. This work paves the way for a wider adoption of tree-decomposition-based algorithms in Bioinformatics.


2021 ◽  
Vol 17 (4) ◽  
pp. 1-40
Author(s):  
Amir Abboud ◽  
Keren Censor-Hillel ◽  
Seri Khoury ◽  
Ami Paz

This article proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget : a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P, which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.


2021 ◽  
Vol 17 (4) ◽  
pp. 1-19
Author(s):  
Xiaoming Sun ◽  
David P. Woodruff ◽  
Guang Yang ◽  
Jialin Zhang

We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 , ⃛ , v q , and observes Mv 1 , ⃛ , Mv q . Here the v i can be randomized as well as chosen adaptively as a function of Mv 1 , ⃛ , Mv i-1 . Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.


Algorithmica ◽  
2021 ◽  
Author(s):  
Massimo Cairo ◽  
Shahbaz Khan ◽  
Romeo Rizzi ◽  
Sebastian Schmidt ◽  
Alexandru I. Tomescu

AbstractGiven a directed graph G and a pair of nodes s and t, an s-tbridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of “safety” from bioinformatics. We say that a walk W is safe with respect to a set $${\mathcal {W}}$$ W of s-t walks, if W is a subwalk of all walks in $${\mathcal {W}}$$ W . We start by considering the maximal safe walks when $${\mathcal {W}}$$ W consists of: all s-t paths, all s-t trails, or all s-t walks of G. We show that the solutions for the first two problems immediately follow from finding all s-t bridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset of visible edges. Here we prove a dichotomy between the s-t paths and s-t trails cases, and the s-t walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of s-tarticulation points (nodes appearing in all s-t paths). We thus obtain the best possible results for natural “safety”-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.


Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 532
Author(s):  
V. Akshay ◽  
H. Philathong ◽  
I. Zacharov ◽  
J. Biamonte

The quantum approximate optimization algorithm (QAOA) has become a cornerstone of contemporary quantum applications development. Here we show that the density of problem constraints versus problem variables acts as a performance indicator. Density is found to correlate strongly with approximation inefficiency for fixed depth QAOA applied to random graph minimization problem instances. Further, the required depth for accurate QAOA solution to graph problem instances scales critically with density. Motivated by Google's recent experimental realization of QAOA, we preform a reanalysis of the reported data reproduced in an ideal noiseless setting. We found that the reported capabilities of instances addressed experimentally by Google, approach a rapid fall-off region in approximation quality experienced beyond intermediate-density. Our findings offer new insight into performance analysis of contemporary quantum optimization algorithms and contradict recent speculation regarding low-depth QAOA performance benefits.


Author(s):  
Naoki Kitamura ◽  
Hirotaka Kitagawa ◽  
Yota Otachi ◽  
Taisuke Izumi

AbstractDistributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of $${\tilde{\Omega }}(\sqrt{n} + D)$$ Ω ~ ( n + D ) rounds for several global problems, where n denotes the number of nodes and D the diameter of the input graph. Because such a lower bound is derived from special “hard-core” instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts was initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. In particular, given a graph class $${\mathcal {C}}$$ C , an f-round algorithm for constructing shortcuts of quality q for any instance in $${\mathcal {C}}$$ C results in $${\tilde{O}}(q + f)$$ O ~ ( q + f ) -round algorithms for solving several fundamental graph problems such as minimum spanning tree and minimum cut, for $${\mathcal {C}}$$ C . The main interest on this line is to identify the graph classes allowing the shortcuts that are efficient in the sense of breaking $${\tilde{O}}(\sqrt{n}+D)$$ O ~ ( n + D ) -round general lower bounds. In this study, we consider the relationship between the quality of low-congestion shortcuts and the following four major graph parameters: doubling dimension, chordality, diameter, and clique-width. The key ingredient of the upper-bound side is a novel shortcut construction technique known as short-hop extension, which might be of independent interest.


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