IA Scholar Query: Bipartite Q-Polynomial Distance-Regular Graphs.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSun, 02 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Codes parameterized by the edges of a bipartite graph with a perfect matching
https://scholar.archive.org/work/scwpeli5q5gjpodfbwfgdrobku
In this paper we study the main characteristics of some evaluation codes parameterized by the edges of a bipartite graph with a perfect matching.Manuel Gonzalez Sarabia, Rafael H. Villarrealwork_scwpeli5q5gjpodfbwfgdrobkuSun, 02 Oct 2022 00:00:00 GMTIsing Model on the Affine Plane
https://scholar.archive.org/work/4k3jhgysevavxktfpw72gzqyn4
We demonstrate that the Ising model on a general triangular graph with 3 distinct couplings K_1,K_2,K_3 corresponds to an affine transformed conformal field theory (CFT). Full conformal invariance of the c= 1/2 minimal CFT is restored by introducing a metric on the lattice through the map sinh(2K_i) = ℓ^*_i/ ℓ_i which relates critical couplings to the ratio of the dual hexagonal and triangular edge lengths. Applied to a 2d toroidal lattice, this provides an exact lattice formulation in the continuum limit to the Ising CFT as a function of the modular parameter. This example can be viewed as a quantum generalization of the finite element method (FEM) applied to the strong coupling CFT at a Wilson-Fisher IR fixed point and suggests a new approach to conformal field theory on curved manifolds based on a synthesis of simplicial geometry and projective geometry on the tangent planes.Richard C. Brower, Evan K. Owenwork_4k3jhgysevavxktfpw72gzqyn4Fri, 30 Sep 2022 00:00:00 GMTOptimal transport methods for combinatorial optimization over two random point sets
https://scholar.archive.org/work/lzgi6js5gvfatmtszdhxg2myre
We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in ℝ^d where the edge cost between two points is given by a p-th power of their Euclidean distance. This includes e.g. the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and d ≥ 3, 1≤ p<d. Previous results were limited to the range p<d/2. Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques.Michael Goldman, Dario Trevisanwork_lzgi6js5gvfatmtszdhxg2myreThu, 29 Sep 2022 00:00:00 GMTTopology and adjunction in promise constraint satisfaction
https://scholar.archive.org/work/tuzqedfdtfd7jd7nvikpmdqpie
The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a c-colouring of a graph that is promised to be k-colourable, where c≥ k. This problem naturally generalises to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems.Andrei Krokhin, Jakub Opršal, Marcin Wrochna, Stanislav Živnýwork_tuzqedfdtfd7jd7nvikpmdqpieThu, 29 Sep 2022 00:00:00 GMTOn the Number of Graphs with a Given Histogram
https://scholar.archive.org/work/oj63gdynqvdvzozt2tvu7hgn2e
Let G be a large (simple, unlabeled) dense graph on n vertices. Suppose that we only know, or can estimate, the empirical distribution of the number of subgraphs F that each vertex in G participates in, for some fixed small graph F. How many other graphs would look essentially the same to us, i.e., would have a similar local structure? In this paper, we derive upper and lower bounds on the number of graphs whose empirical distribution lies close (in the Kolmogorov-Smirnov distance) to that of G. Our bounds are given as solutions to a maximum entropy problem on random graphs of a fixed size k that does not depend on n, under d global density constraints. The bounds are asymptotically close, with a gap that vanishes with d at a rate that depends on the concentration function of the center of the Kolmogorov-Smirnov ball.Shahar Stein Ioushua, Ofer Shayevitzwork_oj63gdynqvdvzozt2tvu7hgn2eWed, 28 Sep 2022 00:00:00 GMTNetwork change point localisation under local differential privacy
https://scholar.archive.org/work/imlcqisgkzfflivko5l3rytlmq
Network data are ubiquitous in our daily life, containing rich but often sensitive information. In this paper, we expand the current static analysis of privatised networks to a dynamic framework by considering a sequence of networks with potential change points. We investigate the fundamental limits in consistently localising change points under both node and edge privacy constraints, demonstrating interesting phase transition in terms of the signal-to-noise ratio condition, accompanied by polynomial-time algorithms. The private signal-to-noise ratio conditions quantify the costs of the privacy for change point localisation problems and exhibit a different scaling in the sparsity parameter compared to the non-private counterparts. Our algorithms are shown to be optimal under the edge LDP constraint up to log factors. Under node LDP constraint, a gap exists between our upper bound and lower bound and we leave it as an interesting open problem.Mengchu Li, Thomas B. Berrett, Yi Yuwork_imlcqisgkzfflivko5l3rytlmqWed, 28 Sep 2022 00:00:00 GMTCompressing network populations with modal networks reveals structural diversity
https://scholar.archive.org/work/tkfiq2ubwvgmxlenlkn37imxzy
Analyzing relational data collected over time requires a critical decision: Is one network representation sufficient? Or are more networks needed to capture changing structures? While the choice may be evident in some cases, for example when analyzing a physical system going through abrupt changes between two known states, other datasets can pose more difficult modeling challenges. Here we describe efficient nonparametric methods derived from the minimum description length principle to construct these network representations automatically. The methods input a population of networks measured on the same set of nodes and output a small set of representative networks together with an assignment of each measurement to one of these representative networks. We show that these methods recover planted heterogeneity in synthetic network populations and effectively identify important structural heterogeneities in example network populations representing global trade and the fossil record.Alec Kirkley, Alexis Rojas, Martin Rosvall, Jean-Gabriel Youngwork_tkfiq2ubwvgmxlenlkn37imxzyWed, 28 Sep 2022 00:00:00 GMTThe Terwilliger algebra of the doubled Odd graph
https://scholar.archive.org/work/mf7navmgh5a4nlcryy5a3mkoae
Let 2.O_m+1 denote the doubled Odd graph with vertex set X on a set of cardinality 2m+1, where m≥ 1. Fix a vertex x_0∈ X. Let 𝒜:=𝒜(x_0) denote the centralizer algebra of the stabilizer of x_0 in the automorphism group of 2.O_m+1, and T:=T(x_0) the Terwilliger algebra of 2.O_m+1. In this paper, we first give a basis of 𝒜 by considering the action of the stabilizer of x_0 on X× X and determine the dimension of 𝒜. Furthermore, we give three subalgebras of 𝒜 such that their direct sum is 𝒜 as vector space. Next, for m≥ 3 we find all isomorphism classes of irreducible T-modules to display the decomposition of T in a block-diagonalization form. Finally, we show that the two algebras 𝒜 and T coincide. This result tells us that the graph 2.O_m+1 may be the first example of bipartite but not Q-polynomial distance-transitive graph for which the corresponding centralizer algebra and Terwilliger algebra are equal.Hou Lihang, Gao Suogang, Kang Na, Hou Bowork_mf7navmgh5a4nlcryy5a3mkoaeWed, 28 Sep 2022 00:00:00 GMTUnique Games hardness of Quantum Max-Cut, and a conjectured vector-valued Borell's inequality
https://scholar.archive.org/work/skytuoytojh4tku23vzh2smgum
The Gaussian noise stability of a function f:ℝ^n →{-1, 1} is the expected value of f(x) · f(y) over ρ-correlated Gaussian random variables x and y. Borell's inequality states that for -1 ≤ρ≤ 0, this is minimized by the halfspace f(x) = sign(x_1). In this work, we generalize this result to hold for functions f:ℝ^n → S^k-1 which output k-dimensional unit vectors. Our main conjecture, which we call the vector-valued Borell's inequality, asserts that the expected value of ⟨ f(x), f(y)⟩ is minimized by the function f(x) = x_≤ k / ‖ x_≤ k‖, where x_≤ k = (x_1, ..., x_k). We give several pieces of evidence in favor of this conjecture, including a proof that it does indeed hold in the special case of n = k. As an application of this conjecture, we show that it implies several hardness of approximation results for a special case of the local Hamiltonian problem related to the anti-ferromagnetic Heisenberg model known as Quantum Max-Cut. This can be viewed as a natural quantum analogue of the classical Max-Cut problem and has been proposed as a useful testbed for developing algorithms. We show the following, assuming our conjecture: (1) The integrality gap of the basic SDP is 0.498, matching an existing rounding algorithm. Combined with existing results, this shows that the basic SDP does not achieve the optimal approximation ratio. (2) It is Unique Games-hard (UG-hard) to compute a (0.956+ε)-approximation to the value of the best product state, matching an existing approximation algorithm. (3) It is UG-hard to compute a (0.956+ε)-approximation to the value of the best (possibly entangled) state.Yeongwoo Hwang, Joe Neeman, Ojas Parekh, Kevin Thompson, John Wrightwork_skytuoytojh4tku23vzh2smgumWed, 28 Sep 2022 00:00:00 GMTEfficient Quantum State Tomography with Mode-assisted Training
https://scholar.archive.org/work/hydarwgjebgihi23oosk4a3v7a
Neural networks (NNs) representing quantum states are typically trained using Markov chain Monte Carlo based methods. However, unless specifically designed, such samplers only consist of local moves, making the slow-mixing problem prominent even for extremely simple quantum states. Here, we propose to use mode-assisted training that provides global information via the modes of the NN distribution. Applied to quantum state tomography using restricted Boltzmann machines, this method improves the quality of reconstructed quantum states by orders of magnitude. The method is applicable to other types of NNs and may efficiently tackle problems previously unmanageable.Yuan-Hang Zhang, Massimiliano Di Ventrawork_hydarwgjebgihi23oosk4a3v7aTue, 27 Sep 2022 00:00:00 GMTGraph theoretic and algorithmic aspect of the equitable coloring problem in block graphs
https://scholar.archive.org/work/vqsfhsyt5nd5ff43hm6nlroij4
An equitable coloring of a graph G=(V,E) is a (proper) vertex-coloring of G, such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.Hanna Furmańczyk, Vahan Mkrtchyanwork_vqsfhsyt5nd5ff43hm6nlroij4Tue, 27 Sep 2022 00:00:00 GMTA Survey on Graph Neural Networks and Graph Transformers in Computer Vision: A Task-Oriented Perspective
https://scholar.archive.org/work/hrto4mikbnfltaqwtehpjnojky
Graph Neural Networks (GNNs) have gained momentum in graph representation learning and boosted the state of the art in a variety of areas, such as data mining (e.g., social network analysis and recommender systems), computer vision (e.g., object detection and point cloud learning), and natural language processing (e.g., relation extraction and sequence learning), to name a few. With the emergence of Transformers in natural language processing and computer vision, graph Transformers embed a graph structure into the Transformer architecture to overcome the limitations of local neighborhood aggregation while avoiding strict structural inductive biases. In this paper, we present a comprehensive review of GNNs and graph Transformers in computer vision from a task-oriented perspective. Specifically, we divide their applications in computer vision into five categories according to the modality of input data, i.e., 2D natural images, videos, 3D data, vision + language, and medical images. In each category, we further divide the applications according to a set of vision tasks. Such a task-oriented taxonomy allows us to examine how each task is tackled by different GNN-based approaches and how well these approaches perform. Based on the necessary preliminaries, we provide the definitions and challenges of the tasks, in-depth coverage of the representative approaches, as well as discussions regarding insights, limitations, and future directions.Chaoqi Chen, Yushuang Wu, Qiyuan Dai, Hong-Yu Zhou, Mutian Xu, Sibei Yang, Xiaoguang Han, Yizhou Yuwork_hrto4mikbnfltaqwtehpjnojkyTue, 27 Sep 2022 00:00:00 GMTHomological properties of binomial edge ideal of a graphs
https://scholar.archive.org/work/mbh52qcnsnfeljvc2wfsi4wou4
In this article, we give a comprehensive survey of the recent progress of research on binomial edge ideal of a graph since 2018.Himadri Mukherjee, Priya Daswork_mbh52qcnsnfeljvc2wfsi4wou4Tue, 27 Sep 2022 00:00:00 GMTApproximating Highly Inapproximable Problems on Graphs of Bounded Twin-Width
https://scholar.archive.org/work/5cng63n2f5b3pkf7zifaqzx45y
For any ε > 0, we give a polynomial-time n^ε-approximation algorithm for Max Independent Set in graphs of bounded twin-width given with an O(1)-sequence. This result is derived from the following time-approximation trade-off: We establish an O(1)^2^q-1-approximation algorithm running in time exp(O_q(n^2^-q)), for every integer q ⩾ 0. Guided by the same framework, we obtain similar approximation algorithms for Min Coloring and Max Induced Matching. In general graphs, all these problems are known to be highly inapproximable: for any ε > 0, a polynomial-time n^1-ε-approximation for any of them would imply that P=NP [Hastad, FOCS '96; Zuckerman, ToC '07; Chalermsook et al., SODA '13]. We generalize the algorithms for Max Independent Set and Max Induced Matching to the independent (induced) packing of any fixed connected graph H. In contrast, we show that such approximation guarantees on graphs of bounded twin-width given with an O(1)-sequence are very unlikely for Min Independent Dominating Set, and somewhat unlikely for Longest Path and Longest Induced Path. Regarding the existence of better approximation algorithms, there is a (very) light evidence that the obtained approximation factor of n^ε for Max Independent Set may be best possible. This is the first in-depth study of the approximability of problems in graphs of bounded twin-width. Prior to this paper, essentially the only such result was a polynomial-time O(1)-approximation algorithm for Min Dominating Set [Bonnet et al., ICALP '21].Pierre Bergé, Édouard Bonnet, Hugues Déprés, Rémi Watrigantwork_5cng63n2f5b3pkf7zifaqzx45ySun, 25 Sep 2022 00:00:00 GMTTree decompositions with bounded independence number: beyond independent sets
https://scholar.archive.org/work/jag2dkw3svfxzlxnnh3dx34udy
We continue the study of graph classes in which the treewidth can only be large due to the presence of a large clique, and, more specifically, of graph classes with bounded tree-independence number. In [Dallard, Milanič, and Štorgel, Treewidth versus clique number. II. Tree-independence number, 2022], it was shown that the Maximum Weight Independent Packing problem, which is a common generalization of the Independent Set and Induced Matching problems, can be solved in polynomial time provided that the input graph is given along with a tree decomposition with bounded independence number. We provide further examples of algorithmic problems that can be solved in polynomial time under this assumption. This includes, for all even positive integers d, the problem of packing subgraphs at distance at least d (generalizing the Maximum Weight Independent Packing problem) and the problem of finding a large induced sparse subgraph satisfying an arbitrary but fixed property expressible in counting monadic second-order logic. As part of our approach, we generalize some classical results on powers of chordal graphs to the context of general graphs and their tree-independence numbers.Martin Milanič, Paweł Rzążewskiwork_jag2dkw3svfxzlxnnh3dx34udySun, 25 Sep 2022 00:00:00 GMTTwin-width V: linear minors, modular counting, and matrix multiplication
https://scholar.archive.org/work/t33m3xdoyfee3hoemm273bte4q
We continue developing the theory around the twin-width of totally ordered binary structures, initiated in the previous paper of the series. We first introduce the notion of parity and linear minors of a matrix, which consists of iteratively replacing consecutive rows or consecutive columns with a linear combination of them. We show that a matrix class has bounded twin-width if and only if its linear-minor closure does not contain all matrices. We observe that the fixed-parameter tractable algorithm for first-order model checking on structures given with an O(1)-sequence (certificate of bounded twin-width) and the fact that first-order transductions of bounded twin-width classes have bounded twin-width, both established in Twin-width I, extend to first-order logic with modular counting quantifiers. We make explicit a win-win argument obtained as a by-product of Twin-width IV, and somewhat similar to bidimensionality, that we call rank-bidimensionality. Armed with the above-mentioned extension to modular counting, we show that the twin-width of the product of two conformal matrices A, B over a finite field is bounded by a function of the twin-width of A, of B, and of the size of the field. Furthermore, if A and B are n × n matrices of twin-width d over 𝔽_q, we show that AB can be computed in time O_d,q(n^2 log n). We finally present an ad hoc algorithm to efficiently multiply two matrices of bounded twin-width, with a single-exponential dependence in the twin-width bound: If the inputs are given in a compact tree-like form, called twin-decomposition (of width d), then two n × n matrices A, B over 𝔽_2, a twin-decomposition of AB with width 2^d+o(d) can be computed in time 4^d+o(d)n (resp. 4^d+o(d)n^1+ε), and entries queried in doubly-logarithmic (resp. constant) time.Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Stéphan Thomasséwork_t33m3xdoyfee3hoemm273bte4qSat, 24 Sep 2022 00:00:00 GMTImproved Distributed Network Decomposition, Hitting Sets, and Spanners, via Derandomization
https://scholar.archive.org/work/lnohfvsjgbdirlx37d5cq6dbcu
This paper presents significantly improved deterministic algorithms for some of the key problems in the area of distributed graph algorithms, including network decomposition, hitting sets, and spanners. As the main ingredient in these results, we develop novel randomized distributed algorithms that we can analyze using only pairwise independence, and we can thus derandomize efficiently. As our most prominent end-result, we obtain a deterministic construction for O(log n)-color O(log n ·logloglog n)-strong diameter network decomposition in Õ(log^3 n) rounds. This is the first construction that achieves almost log n in both parameters, and it improves on a recent line of exciting progress on deterministic distributed network decompositions [Rozhoň, Ghaffari STOC'20; Ghaffari, Grunau, Rozhoň SODA'21; Chang, Ghaffari PODC'21; Elkin, Haeupler, Rozhoň, Grunau FOCS'22].Mohsen Ghaffari, Christoph Grunau, Bernhard Haeupler, Saeed Ilchi, Václav Rozhoňwork_lnohfvsjgbdirlx37d5cq6dbcuFri, 23 Sep 2022 00:00:00 GMTCombinatorial optimization and reasoning with graph neural networks
https://scholar.archive.org/work/dszclpgdgfgzrnd562tfbceni4
Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have focused on solving problem instances in isolation, ignoring that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning, especially graph neural networks (GNNs), as a key building block for combinatorial tasks, either directly as solvers or by enhancing exact solvers. The inductive bias of GNNs effectively encodes combinatorial and relational input due to their invariance to permutations and awareness of input sparsity. This paper presents a conceptual review of recent key advancements in this emerging field, aiming at optimization and machine learning researchers.Quentin Cappart, Didier Chételat, Elias Khalil, Andrea Lodi, Christopher Morris, Petar Veličkovićwork_dszclpgdgfgzrnd562tfbceni4Fri, 23 Sep 2022 00:00:00 GMTLocal Distributed Rounding: Generalized to MIS, Matching, Set Cover, and Beyond
https://scholar.archive.org/work/rkpzdjoeg5cydcveey4imjqgxu
We develop a general deterministic distributed method for locally rounding fractional solutions of graph problems for which the analysis can be broken down into analyzing pairs of vertices. Roughly speaking, the method can transform fractional/probabilistic label assignments of the vertices into integral/deterministic label assignments for the vertices, while approximately preserving a potential function that is a linear combination of functions, each of which depends on at most two vertices (subject to some conditions usually satisfied in pairwise analyses). The method unifies and significantly generalizes prior work on deterministic local rounding techniques [Ghaffari, Kuhn FOCS'21; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Fischer DISC'17] to obtain polylogarithmic-time deterministic distributed solutions for combinatorial graph problems. Our general rounding result enables us to locally and efficiently derandomize a range of distributed algorithms for local graph problems, including maximal independent set (MIS), maximum-weight independent set approximation, and minimum-cost set cover approximation. As a highlight, we in particular obtain a deterministic O(log^2Δ·log n)-round algorithm for computing an MIS in the LOCAL model and an almost as efficient O(log^2Δ·loglogΔ·log n)-round deterministic MIS algorithm in the CONGEST model. As a result, the best known deterministic distributed time complexity of the four most widely studied distributed symmetry breaking problems (MIS, maximal matching, (Δ+1)-vertex coloring, and (2Δ-1)-edge coloring) is now O(log^2Δ·log n). Our new MIS algorithm is also the first direct polylogarithmic-time deterministic distributed MIS algorithm, which is not based on network decomposition.Salwa Faour, Mohsen Ghaffari, Christoph Grunau, Fabian Kuhn, Václav Rozhoňwork_rkpzdjoeg5cydcveey4imjqgxuFri, 23 Sep 2022 00:00:00 GMTHigh-order Multi-view Clustering for Generic Data
https://scholar.archive.org/work/tyvq5ssorvdjlbrwpikhahih3e
Graph-based multi-view clustering has achieved better performance than most non-graph approaches. However, in many real-world scenarios, the graph structure of data is not given or the quality of initial graph is poor. Additionally, existing methods largely neglect the high-order neighborhood information that characterizes complex intrinsic interactions. To tackle these problems, we introduce an approach called high-order multi-view clustering (HMvC) to explore the topology structure information of generic data. Firstly, graph filtering is applied to encode structure information, which unifies the processing of attributed graph data and non-graph data in a single framework. Secondly, up to infinity-order intrinsic relationships are exploited to enrich the learned graph. Thirdly, to explore the consistent and complementary information of various views, an adaptive graph fusion mechanism is proposed to achieve a consensus graph. Comprehensive experimental results on both non-graph and attributed graph data show the superior performance of our method with respect to various state-of-the-art techniques, including some deep learning methods.Erlin Pan, Zhao Kangwork_tyvq5ssorvdjlbrwpikhahih3eThu, 22 Sep 2022 00:00:00 GMT