Graph Problems Research Articles

Degree-Constrained Minimum Spanning Hierarchies in Graphs

The minimum spanning tree problem in graphs under budget-type degree constraints (DCMST) is a well-known NP-hard problem. Spanning trees do not always exist, and the optimum can not be approximated within a constant factor. Recently, solutions have been proposed to solve degree-constrained spanning problems in the case of limited momentary capacities of the nodes. For a given node, the constraint represents a limited degree of the node for each visit. Finding the solution with minimum cost is NP-hard and the related algorithms are not trivial. This paper focuses on this new spanning problem with heterogeneous capacity-like degree bounds. The minimum cost solution corresponds to a graph-related structure, i.e., a hierarchy. We study the conditions of its existence, and we propose its exact computation, a heuristic algorithm, and its approximation.

Polynomial Time Algorithm for Shortest Paths in Interval Temporal Graphs

We develop a polynomial time algorithm for the single-source all destinations shortest paths problem for interval temporal graphs (ITGs). While a polynomial time algorithm for this problem is known for contact sequence temporal graphs (CSGs), no such prior algorithm is known for ITGs. We benchmark our ITG algorithm against that for CSGs using datasets that can be solved using either algorithm. Using synthetic datasets, experimentally, we show that our algorithm for ITGs obtains a speedup of up to 32.5 relative to the state-of-the-art algorithm for CSGs.

Dirac points and inverse problems of quantum graphs associated with Archimedean tilings

Abstract One interesting phenomenon of graphene is the presence of the conical singularity
or Dirac points. Using the quantum graph model, we show that there exist three
classes of possible Dirac points for all of the periodic quantum graphs associated with
Archimedean tilings, when the potentials are identical and even. They occur at the
periodic eigenvalues, anti-periodic eigenvalues and other double eigenvalues of the dispersion
relations respectively. We also characterize their associated potentials. Moreover,
we show that there are no other possible Dirac points. Finally, we solve an
inverse spectral problem for the potential, given the knowledge of the pure point and
absolutely continuous spectra.

Online coloring of disk graphs

In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of disks with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of them is an application of a b-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online L(2,1)-labeling.

Mutual-visibility and general position in double graphs and in Mycielskians

The general position problem in graphs is to find the largest possible set of vertices with the property that no three of them lie on a common shortest path. The mutual-visibility problem in graphs is to find the maximum number of vertices that can be selected such that every pair of vertices in the collection has a shortest path between them with no vertex from the collection as an internal vertex. Here, the general position problem and the mutual-visibility problem are investigated in double graphs and in Mycielskian graphs. Sharp general bounds are proved, in particular involving the total and the outer mutual-visibility number of base graphs. Several exact values are also determined, in particular the mutual-visibility number of the double graphs and of the Mycielskian of cycles.

On the Energy Behaviors of the Bellman–Ford and Dijkstra Algorithms: A Detailed Empirical Study

The Single-Source Shortest Paths (SSSP) graph problem is a fundamental computation. This study attempted to characterize concretely the energy behaviors of the two primary methods to solve it, the Bellman–Ford and Dijkstra algorithms. The very different interactions of the algorithms with the hardware may have significant implications for energy. The study was motivated by the multidisciplinary nature of the problem. Gaining better insights should help vital applications in many domains. The work used reliable embedded sensors in an HPC-class CPU to collect empirical data for a wide range of sizes for two graph cases: complete as an upper-bound case and moderately dense. The findings confirmed that Dijkstra’s algorithm is drastically more energy efficient, as expected from its decisive time complexity advantage. In terms of power draw, however, Bellman–Ford had an advantage for sizes that fit in the upper parts of the memory hierarchy (up to 2.36 W on average), with a region of near parity in both power draw and total energy budgets. This result correlated with the interaction of lighter logic and graph footprint in memory with the Level 2 cache. It should be significant for applications that rely on solving a lot of small instances since Bellman–Ford is more general and is easier to implement. It also suggests implications for the design and parallelization of the algorithms when efficiency in power draw is in mind.

Mutual-visibility problems in Kneser and Johnson graphs

Mutual-visibility problems in Kneser and Johnson graphs

Learning bipartite graphs from spectral templates

Graph learning is crucial for understanding the relationship between data components. Signal processing-based graph learning algorithms are designed for specific signal models. This work investigates the problem of learning bipartite graphs given arbitrarily ordered spectral templates or graph eigenvectors. Starting from the spectral templates, the proposed algorithm identifies the vertex groups of the bipartite graph. Experiments conducted on three different types of synthetic datasets demonstrate that the proposed bipartite graph learning algorithms outperform structure-blind learning techniques across various signal-to-noise (SNR) regimes. Our algorithm leverages the spectral signatures of a bipartite graph, specifically the structure of the graph’s eigenvectors.

Bipartite unicyclic graphs with a unique perfect matching having the smallest positive eigenvalue equal to

The smallest positive eigenvalue τ ( G ) of a simple graph G is the smallest positive eigenvalue of its adjacency matrix A ( G ) . In [F. J. Zhang and A. Chang, Acyclic molecules with greatest HOMO-LUMO separation, Discrete Applied Mathematics, 98:165–171, (1999).], the authors characterized all nonsingular trees with τ equal to 2 − 1 . We consider the same problem for bipartite unicyclic graphs with a unique perfect matching. Let U be the class of all connected bipartite unicyclic graphs with a unique perfect matching. In this article, we characterize all graphs U in U with the property that τ ( U ) = 2 − 1 . Further, we show that the largest limit point of the smallest positive eigenvalues of graphs in U is 2 − 1 , whereas the smallest limit point is 0.

Some Algorithmic Results for Eternal Vertex Cover Problem in Graphs

The eternal vertex cover problem is a variant of the vertex cover problem. It is a two-player (attacker and defender) game in which, given a graph $G=(V,E)$, the defender needs to allocate guards at some vertices so that the allocated vertices form a vertex cover. The attacker can attack one edge at a time, and the defender needs to move the guards along the edges such that at least one guard moves through the attacked edge and the new configuration still remains a vertex cover. The attacker wins if no such move exists for the defender. The defender wins if there exists a strategy to defend the graph against infinite sequence of attacks. The minimum number of guards with which the defender can form a winning strategy is called the eternal vertex cover number of $G$, and is denoted by $evc(G)$. Given a graph $G$, the problem of finding the eternal vertex cover number is NP-hard for general graphs and remains NP-hard even for bipartite graphs. We have given a polynomial time algorithm to find the Eternal vertex cover number in chain graphs and $P_4$-sparse graphs. We have also given a linear-time algorithm to find the eternal vertex cover number for split graphs, an important subclass of chordal graphs.

NeRF-Texture: Synthesizing Neural Radiance Field Textures.

Texture synthesis is a fundamental problem in computer graphics that would benefit various applications. Existing methods are effective in handling 2D image textures. In contrast, many real-world textures contain meso-structure in the 3D geometry space, such as grass, leaves, and fabrics, which cannot be effectively modeled using only 2D image textures. We propose a novel texture synthesis method with Neural Radiance Fields (NeRF) to capture and synthesize textures from given multi-view images. In the proposed NeRF texture representation, a scene with fine geometric details is disentangled into the meso-structure textures and the underlying base shape. This allows textures with meso-structure to be effectively learned as latent features situated on the base shape, which are fed into a NeRF decoder trained simultaneously to represent the rich view-dependent appearance. Using this implicit representation, we can synthesize NeRF-based textures through patch matching of latent features. However, inconsistencies between the metrics of the reconstructed content space and the latent feature space may compromise the synthesis quality. To enhance matching performance, we further regularize the distribution of latent features by incorporating a clustering constraint. In addition to generating NeRF textures over a planar domain, our method can also synthesize NeRF textures over curved surfaces, which are practically useful. Experimental results and evaluations demonstrate the effectiveness of our approach.

Reconfiguring Shortest Paths in Graphs

Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming P≠NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\ exttt {P}}\\,}}\ e {{\\,\\mathrm{\ exttt {NP}}\\,}}$$\\end{document}), even for relaxed variants of the problem (assuming P≠PSPACE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\ exttt {P}}\\,}}\ e {{\\,\\mathrm{\ exttt {PSPACE}}\\,}}$$\\end{document}). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 2$$\\end{document}) contiguous vertices on a shortest path can be changed at a time.

On non-superperfection of edge intersection graphs of paths

The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.

Semantic-aware hyper-space deformable neural radiance fields for facial avatar reconstruction

High-fidelity facial avatar reconstruction from monocular videos is a prominent research problem in computer graphics and computer vision. Recent advancements in the Neural Radiance Field (NeRF) have demonstrated remarkable proficiency in rendering novel views and garnered attention for its potential in facial avatar reconstruction. However, previous methodologies have overlooked the complex motion dynamics present across the head, torso, and intricate facial features. Additionally, a deficiency exists in a generalized NeRF-based framework for facial avatar reconstruction adaptable to either 3DMM coefficients or audio input. To tackle these challenges, we propose an innovative framework that leverages semantic-aware hyper-space deformable NeRF, facilitating the reconstruction of high-fidelity facial avatars from either 3DMM coefficients or audio features. Our framework effectively addresses both localized facial movements and broader head and torso motions through semantic guidance and a unified hyper-space deformation module. Specifically, we adopt a dynamic weighted ray sampling strategy to allocate varying degrees of attention to distinct semantic regions, enhancing the deformable NeRF framework with semantic guidance to capture fine-grained details across diverse facial regions. Moreover, we introduce a hyper-space deformation module that enables the transformation of observation space coordinates into canonical hyper-space coordinates, allowing for the learning of natural facial deformation and head-torso movements. Extensive experiments validate the superiority of our framework over existing state-of-the-art methods, demonstrating its effectiveness in producing realistic and expressive facial avatars. Our code is available at https://github.com/jematy/SAHS-Deformable-Nerf.

Temporal subgraph matching method for multi-connected temporal graph

Temporal graph matching is used to find a matching subgraph set that satisfies temporal and topological information on the temporal graph. It is widely used in real scenarios and provides important support for big data mining and analysis. Most existing matching methods take the single-connected graph as the matching graph, which has a high computational cost. Simultaneously, the construction and matching complexity of its index on multi-connected temporal subgraph matching is high. To solve the above problems, we propose a new temporal subgraph matching problem for multi-connected temporal graphs and proposes the corresponding efficient filtering-selecting-matching three-stage multi-connected temporal subgraph matching method. In the filtering stage, a vertex-based filter is proposed to generate a candidate temporal subgraph set. In the select phase, a set of temporal subgraphs satisfying the vertex mapping is found. In the matching stage, the single-filter temporal subgraph matching method and temporal subgraph matching based on the temporal information tree index method are proposed to solve the problems that the edges of multi-connected temporal subgraphs cannot be mapped and the single graph matching efficiency is low, respectively. The experimental results show that the proposed method improves the matching accuracy, and the corresponding index size is reduced.

Structure and position-aware graph neural network for airway labeling

We present a novel graph-based approach for labeling the anatomical branches of a given airway tree segmentation. The proposed method formulates airway labeling as a branch classification problem in the airway tree graph, where branch features are extracted using convolutional neural networks and enriched using graph neural networks. Our graph neural network is structure-aware by having each node aggregate information from its local neighbors and position-aware by encoding node positions in the graph.We evaluated the proposed method on 220 airway trees from subjects with various severity stages of Chronic Obstructive Pulmonary Disease (COPD). The results demonstrate that our approach is computationally efficient and significantly improves branch classification performance than the baseline method. The overall average accuracy of our method reaches 91.18% for labeling 18 segmental airway branches, compared to 83.83% obtained by the standard CNN method and 87.37% obtained by the existing method. Furthermore, the reader study done on an additional set of 40 subjects shows that our algorithm performs comparably to human experts in labeling segmental-airways. We published our source code at https://github.com/DIAGNijmegen/spgnn. The proposed algorithm is also publicly available at https://grand-challenge.org/algorithms/airway-anatomical-labeling/.

Pose-Aware Attention Network for Flexible Motion Retargeting by Body Part.

Motion retargeting is a fundamental problem in computer graphics and computer vision. Existing approaches usually have many strict requirements, such as the source-target skeletons needing to have the same number of joints or share the same topology. To tackle this problem, we note that skeletons with different structure may have some common body parts despite the differences in joint numbers. Following this observation, we propose a novel, flexible motion retargeting framework. The key idea of our method is to regard the body part as the basic retargeting unit rather than directly retargeting the whole body motion. To enhance the spatial modeling capability of the motion encoder, we introduce a pose-aware attention network (PAN) in the motion encoding phase. The PAN is pose-aware since it can dynamically predict the joint weights within each body part based on the input pose, and then construct a shared latent space for each body part by feature pooling. Extensive experiments show that our approach can generate better motion retargeting results both qualitatively and quantitatively than state-of-the-art methods. Moreover, we also show that our framework can generate reasonable results even for a more challenging retargeting scenario, like retargeting between bipedal and quadrupedal skeletons because of the body part retargeting strategy and PAN.

Graph learning from incomplete graph signals: From batch to online methods

Inferring graph topologies from data is crucial in many graph-related applications. Existing works typically assume that signals are observed at all nodes, which may not hold due to application-specific constraints. The problem becomes more challenging when data are sequentially available and no delay is tolerated. To address these issues, we propose an approach for learning graphs from incomplete data. First, the problem of learning graphs with missing data is formulated as maximizing the posterior distribution with hidden variables from a Bayesian perspective. Then, we propose an expectation maximization (EM) algorithm to solve the induced problem, in which graph learning and graph signal recovery are jointly performed. Furthermore, we extend the proposed EM algorithm to an online version to accommodate the delay-sensitive situations of sequential data. Theoretically, we analyze the dynamic regret of the proposed online algorithm, illustrating the effectiveness of our algorithm in tracking graphs from partial observations in an online manner. Finally, extensive experiments on synthetic and real data are conducted, and the results corroborate that our approach can learn graphs effectively from incomplete data in both batch and online situations.

Classical computation over quantum architectures

Abstract The lack of purely Quantum Programming Languages constitutes a hurdle in the general description of quantum computational processes; the implementation is heavily dependent on the considered quantum computational model. To bypass the obstacle, this paper pursues a new direction, investigating the compilation of classical programming paradigms over different quantum computational models: Gate-Based, Measurement-Based and Adiabatic Quantum Computation. Since graphs can be exploited to describe both classical and quantum computations, the problem of graph encoding on quantum hardware is tightly connected to our purposes. As such, it holds a major relevance in our quest for quantum compilation. While studying these topics through the lenses of Graph Theory, declarative programming emerges as the ideal candidate for such endeavour. In this paper we consider some existing quantum computational models and for each of them we identify the main subtleties in the compilation of classical languages. In turn, we break these complexities down into easier problems to stimulate further developments in this area of research. As it turns out, the observations for each model differ widely. Nevertheless, as for the tasks here considered, no model seems to claim supremacy over the others. In contrast, declarative programming maintains the spot as the ideal candidate for quantum compilation, independently of the model.

On Three Domination-based Identification Problems in Block Graphs

The problems of determining the minimum-sized identifying, locating-dominating and open locating-dominating codes of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set C of a graph G such that the vertices of a chosen subset of V(G) (i.e. either V(G) \ C or V(G) itself) are uniquely determined by their neighborhoods in C. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graph classes. In this work, we present tight lower and upper bounds for all three types of codes for block graphs (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or blocks) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.