General Undirected Graphs Research Articles

Toward Cross-Domain Dynamic Accumulator Authentication Based on Blockchain in Internet of Things

The authentication problem is one of the most significant challenges in the applications of Internet of Things (IoT), and the relationship authentication among smart devices is an effective solution to tackle cross-domain issue. In this article, we first abstract a general undirected graph from the authentication relationship among the smart devices in IoT. Then, we formulate the authentication problem into a signature transitivity problem by incorporating accumulator knowledge and a standard digital signature scheme. As a consequence, the legality of authentication is well-verified by computing the signature and witness of the related edges without worrying about whether the devices belong to the same administrative domain or not. Finally, the efficiency and online time of authority are solved by exploring the blockchain technology, which also leads to the assurance of an online 24-hour third party. The results of analysis and comparison show that the proposed scheme CroDA can well address the practical authentication issue.

Classification of edges in a general graph associated with the change in multiplicity of an eigenvalue

ABSTRACT We investigate the change in the multiplicities of an eigenvalue of an Hermitian matrix whose graph is a general undirected graph, when an edge is removed from the graph. The same question when the graph is a tree has been investigated in prior work. Here, we give possible classifications of edges in a general graph in terms of the statuses of adjacent vertices. It turns out that there are four cases that do occur in a general graph but cannot occur in a tree.

On generating random Gaussian graphical models

Structure learning methods for covariance and concentration graphs are often validated on synthetic models, usually obtained by randomly generating: (i) an undirected graph, and (ii) a compatible symmetric positive definite (SPD) matrix. In order to ensure positive definiteness in (ii), a dominant diagonal is usually imposed. In this work we investigate different methods to generate random symmetric positive definite matrices with undirected graphical constraints. We show that if the graph is chordal it is possible to sample uniformly from the set of correlation matrices compatible with the graph, while for general undirected graphs we rely on a partial orthogonalization method.

Scaling matrices and counting the perfect matchings in graphs

We investigate efficient randomized methods for approximating the number of perfect matchings in bipartite graphs and general undirected graphs. Our approach is based on assigning probabilities to edges, randomly selecting an edge to be in a perfect matching, and discarding edges that cannot be put in a perfect matching. The probabilities are set according to the entries in the doubly stochastically scaled version of the adjacency matrix of the given graph. The experimental analysis on random and real-life graphs shows improvements in the approximation over previous and similar methods from the literature.

A multi-start local search algorithm for the Hamiltonian completion problem on undirected graphs

This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian completion problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.

The maximum 1-2 matching problem and two kinds of its variants

We present an algorithm for solving the maximum 1-2 matching problem in bipartite graphs by reducing it to solving the 1-1 matching in general undirected graphs. Further, we propose two variants of the maximum 1-2 matching problem: one is the maximum 1-2 matching problem of the different kind in divided bipartite graphs and the other is the maximum 1-2 matching problem of the same kind in divided bipartite graphs. The former one is shown to be solved by a similar algorithm to that of the maximum 1-2 matching problem by reducing it to solve the maximum 1-1 matching problem in bipartite graphs. For the latter one, we can only present the augmenting subgraph theorem, which is an extension of our augmenting trail theorem for the maximum 1-2 matching problem.

Max flow vitality in general and st‐planar graphs

AbstractThe vitality of an arc/node of a graph with respect to the maximum flow between two fixed nodes s and t is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper, we address the issue of determining the vitality of arcs and/or nodes for the maximum flow problem. We show how to compute the vitality of all arcs in a general undirected graph by solving only 2(n − 1) max flow instances and, in st‐planar graphs (directed or undirected) we show how to compute the vitality of all arcs and all nodes in O(n) worst‐case time. Moreover, after determining the vitality of arcs and/or nodes, and given a planar embedding of the graph, we can determine the vitality of a “contiguous” set of arcs/nodes in time proportional to the size of the set.

A Flexible, Real-Time Algorithm for Simulating Correlated Random Fields and Its Properties

Contemporary real-time problems like CAPTCHA generation and optical character recognition can be solved effectively using correlated random fields. These random fields should be produced on a graph in order that problems of any dimension and shape can be handled. However, traditional solutions are often too slow, inaccurate or both. Herein, the Quick Simulation Random Field algorithm to produce correlated random fields on general undirected graphs is introduced. It differs from prior algorithms by completing the graph and setting the unspecified covariances to zero, which facilitates analytic study. The Quick Simulation Random Field graph distribution is derived within and the following questions are studied: (1) For which marginal pmfs and covariances will this algorithm work? (2) When does the marginal property hold, where the sub-graph distribution of an algorithm-simulated field matches the distribution of the algorithm-simulated field on the subgraph? (3) When does the permutation property hold, where the vertex simulation order does not affect the joint distribution?

Space-efficient path-reporting approximate distance oracles

We consider approximate path-reporting distance oracles, distance labeling and labeled routing with extremely low space requirements, for general undirected graphs. For distance oracles, we show how to break the nlog⁡n space bound of Thorup and Zwick if approximate paths rather than distances need to be reported. For approximate distance labeling and labeled routing, we break the previously best known space bound of O(log⁡n) words per vertex. The cost for such space efficiency is an increased stretch.

Approximating the restricted 1-center in graphs

This paper studies the restricted vertex 1-center problem (RV1CP) and restricted absolute 1-center problem (RA1CP) in general undirected graphs with each edge having two weights, cost and delay. First, we devise a simple FPTAS for RV1CP with O(mn3(1ϵ+log⁡log⁡n)) running time, based on FPTAS proposed by Lorenz and Raz (1999) [11] for computing end-to-end restricted shortest path (RSP). During the computation of the FPTAS for RV1CP, we derive a RSP distance matrix. Next, we discuss RA1CP in such graphs where the delay is a separable (e.g., linear) function of the cost on edge. We investigate an important property that the FPTAS for RV1CP can find a (1+ϵ)-approximation of RA1CP when the RSP distance matrix has a saddle point. In addition, we show that it is harder to find an approximation of RA1CP when the matrix has no saddle point. This paper develops a scaling algorithm with at most O(mn3K(log⁡Kη+log⁡log⁡n)) running time where K is a step-size parameter and η is a given positive number, to find a (1+η)-approximation of RA1CP.

Hitting Forbidden Minors: Approximation and Kernelization

We study a general class of problems called $\mathcal{F}$-Deletion problems. In an $\mathcal{F}$-Deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family ${\cal F}$ of forbidden minors. We study the problem parameterized by $k$, using $p$-$\mathcal{F}$-Deletion to refer to the parameterized version of the problem. We obtain a number of algorithmic results on the $p$-$\mathcal{F}$-Deletion problem when $\mathcal{F}$ contains a planar graph. We give a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leaves, as an induced subgraph, where $t$ is a fixed integer and an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when $\cal F$ only contains graph $\theta_c$ as a minor for a fixed integer $c$. The graph $\theta_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set, and Diamond Hitting Set. The generic kernelization algorithm is based on a nontrivial application of protrusion techniques, previously used only for problems on topological graph classes.

Near-optimal communication-time tradeoff in fault-tolerant computation of aggregate functions

This paper considers the problem of computing general commutative and associative aggregate functions (such as Sum) over distributed inputs held by nodes in a distributed system, while tolerating failures. Specifically, there are N nodes in the system, and the topology among them is modeled as a general undirected graph. Whenever a node sends a message, the message is received by all of its neighbors in the graph. Each node has an input, and the goal is for a special root node (e.g., the base station in wireless sensor networks or the gateway node in wireless ad hoc networks) to learn a certain commutative and associate aggregate of all these inputs. All nodes in the system except the root node may experience crash failures, with the total number of edges incidental to failed nodes being upper bounded by f. The timing model is synchronous where protocols proceed in rounds. Within such a context, we focus on the following question: Under any given constraint on time complexity, what is the lowest communication complexity, in terms of the number of bits sent (i.e., locally broadcast) by each node, needed for computing general commutative and associate aggregate functions? This work, for the first time, reduces the gap between the upper bound and the lower bound for the above question from polynomial to polylog. To achieve this reduction, we present significant improvements over both the existing upper bounds and the existing lower bounds on the problem.

Decomposition of Network Communication Games

Using network control structures this paper introduces network communication games as a generalization of vertex games and edge games corresponding to communication situations and studies their decomposition into unanimity games. We obtain a relation between the dividends of the network communication game and the underlying transferable utility game, which depends on the structure of the undirected graph. This relation extends the computational results for tree communication networks to general undirected graphs and is used to derive new characterizations of the Myerson value and the position value. Moreover, network communication games also allow to consider both the vertices and the edges of the graph as players, leading to a new network value.

Global consensus of single‐integrator agents subject to saturation constraints

This paper investigates the problem of leader-following consensus of single-integrator agents subject to saturation constraints. Global leader-following consensus protocol is developed with proper choice of the relative coupling gain. Both the case of input saturation and the case of actuator saturation are considered. It is shown that global consensus of the multiagent system can be reached under the general undirected graph provided that its augmented graph contains a directed spanning tree. Numerical examples are provided to demonstrate the theoretical results.

Deterministic Rendezvous, Treasure Hunts, and Strongly Universal Exploration Sequences

We obtain several improved solutions for the deterministic rendezvous problem in general undirected graphs. Our solutions answer several problems left open by Dessmark et al. We also introduce an interesting variant of the rendezvous problem, which we call the deterministic treasure hunt problem. Both the rendezvous and the treasure hunt problems motivate the study of universal traversal sequences and universal exploration sequences with some strengthened properties. We call such sequences strongly universal traversal (exploration) sequences . We give an explicit construction of strongly universal exploration sequences. The existence of strongly universal traversal sequences, as well as the solution of the most difficult variant of the deterministic treasure hunt problem, are left as intriguing open problems.

Attitude synchronization of satellites with internal actuation

We present control synthesis for attitude synchronization of a cluster of satellites, that are internally actuated, using a coordinate-free approach. The control law is designed using the notion of potential energy shaping that rests on the use of modified trace functions. The distinguishing aspect of the internal actuation model is that the drift vector field in the system dynamics explicitly depends on the conserved total angular momentum. The control law presented is applicable to a general undirected graph of communication between the satellites, with one satellite of the cluster communicating with the leader. Further, the control law is applicable with just two vector observations in each satellite, and hence making redundant, complete attitude information.

An efficient algorithm for exact evaluation of stochastic watersheds

The stochastic watershed is a method for unsupervised image segmentation proposed by Angulo and Jeulin (2007). The method first computes a probability density function (PDF), assigning to each piece of contour in the image the probability to appear as a segmentation boundary in seeded watershed segmentation with randomly selected seeds. Contours that appear with high probability are assumed to be more important. This PDF is then post-processed to obtain a final segmentation. The main computational hurdle with the stochastic watershed method is the calculation of the PDF. In the original publication by Angulo and Jeulin, the PDF was estimated by Monte Carlo simulation, i.e., repeatedly selecting random markers and performing seeded watershed segmentation. Meyer and Stawiaski (2010) showed that the PDF can be calculated exactly, without performing any Monte Carlo simulations, but do not provide any implementation details. In a naive implementation, the computational cost of their method is too high to make it useful in practice. Here, we extend the work of Meyer and Stawiaski by presenting an efficient (quasi-linear) algorithm for exact computation of the PDF. We demonstrate that in practice, the proposed method is faster than any previously reported method by more than two orders of magnitude. The algorithm is formulated for general undirected graphs, and thus trivially generalizes to images with any number of dimensions.

On A New Competitive Measure For Oblivious Routing

Oblivious routing algorithms use only locally avail¬able information at network nodes to forward traffic, and as such, a plausible choice for distributed implementations. It is a natural desire to quantify the performance penalty we pay for this distributedness. Recently, Racke has shown that for general undirected graphs the competitive ratio is only O(log n), that is, the maximum congestion caused by the oblivious algorithm is within a logarithmic factor of the best possible congestion. And while the performance penalty is larger for directed networks (Azar gives a O(vn) lower bound), experiments on many real-world topologies show that it usually remains under 2. These competitive measures, however, are of worst-case type, and therefore do not always give adequate characterization. The more different combinations of demands a routing algorithm can accommodate in the network without congestion, the better. Driven by this observation, in this paper we introduce a new competitive measure, the volumetric competitive ratio, as the measure of all admissible demands compared to the measure of demands routed without congestion. The main result of the paper is a general lower bound on the volumetric ratio; and we also show a directed graph with O(1) competitive ratio that exhibits O(n) volumetric ratio. Our numerical evaluations show that the competitivity of oblivious routing in terms of the new measure quickly vanishes even in relatively small common-place topologies.

Global consensus of multiple integrator agents via saturated controls

This paper investigates the problem of global leader-following consensus of multiple integrator agents subject to control input saturation. A weighted and saturated consensus algorithm is proposed to solve this problem. Both the case of an undirected communication topology and the case of a directed communication topology are considered. It is shown that global consensus of the multiple integrator agents can be reached under a general undirected graph or a detailed balanced directed graph provided that its generated graph contains a directed spanning tree. Numerical examples are provided to demonstrate the theoretical results.

The All-or-Nothing Multicommodity Flow Problem

We consider the all-or-nothing multicommodity flow problem in general graphs. We are given a capacitated undirected graph $G=(V,E,u)$ and a set of $k$ node pairs $s_1 t_1, s_2t_2, \ldots ,s_kt_k$. Each pair has a unit demand. A subset $S$ of $\{1,2,\ldots,k\}$ is routable if there is a multicommodity flow in $G$ that simultaneously sends one unit of flow between $s_i$ and $t_i$ for each $i$ in $S$. Note that this differs from the edge-disjoint path problem (edp) in that we do not insist on integral flows for the pairs. The objective is to find a maximum routable subset $S$. When $G$ is a capacitated tree, the problem already generalizes $b$-matchings, and even in this case it is NP-hard and APX-hard to approximate. For trees, a $2$-approximation is known for the cardinality case and a $4$-approximation for the weighted case. In this paper we show that the natural linear programming relaxation for the all-or-nothing flow problem has a polylogarithmic integrality gap in general undirected graphs. This is in sharp contrast to edp, where the gap is known to be $\Theta(\sqrt{n})$; this ratio is also the best approximation ratio currently known for edp. Our algorithm extends to the case where each pair $s_it_i$ has a demand $d_i$ associated with it and we need to completely route $d_i$ to get credit for pair $i$; we assume that the maximum demand of the pairs is at most the minimum capacity of the edges. We also consider the online admission control version where pairs arrive online and the algorithm has to decide immediately on its arrival whether to accept it and the accepted pairs have to be routed. We obtain a randomized algorithm which has a polylogarithmic competitive ratio for maximizing throughput of the accepted requests if it is allowed to violate edge capacities by a $(2+\epsilon)$ factor.