Extra Edges Research Articles

The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems

Given a directed graph G and a list ( s 1 , t 1 ), …, ( s d , t d ) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed s i → t i path for every 1≤ i ≤ d . The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t 1 , …, t d ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every t i to every other t j ) is known to be W[1]-hard parameterized by the number of terminals. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if ℋ is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list ( s 1 , t 1 ), …, ( s d , t d ) of demands form a directed graph that is a member of ℋ. Our main result is a complete characterization of the classes ℋ resulting in fixed-parameter tractable special cases: we show that if every pattern in ℋ has the combinatorial property of being “transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges,” then the problem is FPT, and it is W[1]-hard for every recursively enumerable ℋ not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], q -Root Steiner Tree is FPT for constant q [Suchý, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [Guo et al., SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before.

A defense strategy for false data injection attacks in multi-agent systems

The issue of defense strategy for cooperative control in multi-agent systems under false data injection attacks is investigated in this paper. False data injection attacks are typical cyber-attacks, some of which can evade detection and result in system instability. A communication strategy is proposed to ensure the security of multi-agent systems. In the proposed defense strategy, each agent is assigned to transmit its state to its neighbours and extra communication edges are stochastically triggered by the designed trigger signals. Furthermore, the communication topology condition is derived to ensure the security of multi-agent systems under the defense strategy. The theoretical results are verified through numerical simulation.

Refined Fuzzy Profile Matching

A profile describes a set of properties, e.g. a set of skills a person may have or a set of skills required for a particular job. Profile matching aims to determine how well a given profile fits to a requested profile and vice versa. Fuzzyness is naturally attached to this problem. The filter-based matching theory uses filters in lattices to represent profiles, and matching values in the interval [0,1], so the lattice order refers to subsumption between the concepts in a profile. In this article the lattice is extended by additional information in form of weighted extra edges that represent partial quantifiable relationships between these concepts. This gives rise to fuzzy filters, which permit a refinement of profile matching. Another way to introduce fuzzyness is to treat profiles as fuzzy sets. In the present paper we combine these two aproaches. Extra edges may introduce directed cycles in the directed graph of the ontology, and the structure of a lattice is lost. We provide a construction grounded in formal concept analysis to extend the original lattice and remove the cycles such that matching values determined over the extended lattice are exactly those resulting from the use of fuzzy filters in case of crisp profiles. For fuzzy profiles we show how to modify the weighting construction while eliminating the directed cycles but still regaining the matching values. We also give sharp estimates for the growth of the number of vertices in this construction.

Redundantly globally rigid braced triangulations

By mapping the vertices of a graph G to points in ℝ3, and its edges to the corresponding line segments, we obtain a three-dimensional realization of G. A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.

Quantum Walk Search on a Two-dimensional Grid with Extra Edges

Quantum Walk Search on a Two-dimensional Grid with Extra Edges

Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches

We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a $k$-edge-connected graph $G=(V,E)$ and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to $G$ makes the graph $(k+1)$-edge-connected. If $k$ is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP) -- i.e., $G$ is a spanning tree -- for which significant progress has been achieved recently, leading to approximation factors below $1.5$ (the currently best factor is $1.458$). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below $2$ for CAP by presenting a $1.91$-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below $1.5$, based on a new analysis technique. Through these ingredients, we obtain a $1.393$-approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned $1.91$-approximation for CAP and also the previously best approximation factor of $1.458$ for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below $1.5$.

Regular Splitting Graph Network for 3D Human Pose Estimation.

In human pose estimation methods based on graph convolutional architectures, the human skeleton is usually modeled as an undirected graph whose nodes are body joints and edges are connections between neighboring joints. However, most of these methods tend to focus on learning relationships between body joints of the skeleton using first-order neighbors, ignoring higher-order neighbors and hence limiting their ability to exploit relationships between distant joints. In this paper, we introduce a higher-order regular splitting graph network (RS-Net) for 2D-to-3D human pose estimation using matrix splitting in conjunction with weight and adjacency modulation. The core idea is to capture long-range dependencies between body joints using multi-hop neighborhoods and also to learn different modulation vectors for different body joints as well as a modulation matrix added to the adjacency matrix associated to the skeleton. This learnable modulation matrix helps adjust the graph structure by adding extra graph edges in an effort to learn additional connections between body joints. Instead of using a shared weight matrix for all neighboring body joints, the proposed RS-Net model applies weight unsharing before aggregating the feature vectors associated to the joints in order to capture the different relations between them. Experiments and ablations studies performed on two benchmark datasets demonstrate the effectiveness of our model, achieving superior performance over recent state-of-the-art methods for 3D human pose estimation.

3D Shape Deformation Using Stick Figures

Obtaining new poses of an articulated character is a critical task in computer graphics. We address this issue with a new shape deformation approach consisting of two phases enabling the user to express the new pose as a simple stick figure, also called skeleton consisting of few bones. We interchangeably refer to this problem as shape transfer because the template shape is transferred to the independent stick figure that may be obtained from any source such as motion capture and 3D sketching. In the first phase, the stick figure is embedded into the template shape, resembling the rigging process of character skinning. Then, instead of computing blend weights as in skinning, the shape, discretized as a mesh, is augmented by adding extra edges between a subset of mesh vertices and the embedded stick figure. The second phase deforms the augmented mesh towards the new pose under the guidance of the embedded stick figure by minimizing an As-Rigid-As-Possible (ARAP) energy. The overall deformation is intuitive as skinning, preserves surface details as it is based on ARAP deformation yet has better volume preservation capability owing to the augmented mesh. Our results are validated both in terms of timing and accuracy in a comprehensive test suite that includes state-of-the-art deformation techniques. • Edges between mesh and its skeleton prevent from volume loss in ARAP deformation. • Two skeletons can be matched by minimizing a distance-based penalty function. • Distance-based skeleton matching method can be used as an automatic rigging method. • New poses of a shape can be created by using a stick figure and a template mesh.

AEDNet: Adaptive Edge-Deleting Network For Subgraph Matching

Subgraph matching is to find all subgraphs in a data graph that are isomorphic to an existing query graph. Subgraph matching is an NP-hard problem, yet has found its applications in many areas. Many learning-based methods have been proposed for graph matching, whereas few have been designed for subgraph matching. The subgraph matching problem is generally more challenging, mainly due to the different sizes between the two graphs, resulting in considerable large space of solutions. Also the extra edges existing in the data graph connecting to the matched nodes may lead to two matched nodes of two graphs having different adjacency structures and often being identified as distinct objects. Due to the extra edges, the existing learning based methods often fail to generate sufficiently similar node-level embeddings for matched nodes. This study proposes a novel Adaptive Edge-Deleting Network (AEDNet) for subgraph matching. The proposed method is trained in an end-to-end fashion. In AEDNet, a novel sample-wise adaptive edge-deleting mechanism removes extra edges to ensure consistency of adjacency structure of matched nodes, while a unidirectional cross-propagation mechanism ensures consistency of features of matched nodes. We applied the proposed method on six datasets with graph sizes varying from 20 to 2300. Our evaluations on six open datasets demonstrate that the proposed AEDNet outperforms six state-of-the-arts and is much faster than the exact methods on large graphs.

A graph-based path planning method for additive manufacturing of continuous fiber-reinforced planar thin-walled cellular structures

PurposeComposite cellular structures have wide application in advanced engineering fields due to their high specific stiffness and strength. As an emerging technology, continuous fiber-reinforced polymer additive manufacturing provides a cost-effective solution for fabricating composite cellular structures with complex designs. However, the corresponding path planning methods are case-specific and have not considered any manufacturing constraints. This study aims to develop a generally applicable path planning method to fill the above research gap.Design/methodology/approachThis study proposes a path planning method based on the graph theory, yielding an infill toolpath with a minimum fiber cutting frequency, printing time and total turning angle. More specifically, the cellular structure design is converted to a graph first. Then, the graph is modified to search an Eulerian path by adding an optimal set of extra edges determined through the integer linear programming method. Finally, the toolpath with minimum total turning angle is obtained with a constrained Euler path search algorithm.FindingsThe effectiveness of the proposed method is validated through the fabrication of both periodic and nonperiodic composite cellular structures, i.e. triangular unit cell-based, Voronoi diagram-based and topology optimized structures. The proposed method provides the basis for manufacturing planar thin-walled cellular structures of continuous fiber-reinforced polymer (CFRP). Moreover, the proposed method shows a notable improvement compared with the existing method. The fiber cutting frequency, printing time and total turning angle have been reduced up to 88.7%, 52.6% and 65.5%, respectively.Originality/valueA generally applicable path planning method is developed to generate continuous toolpaths for fabricating cellular structures in CFRP-additive manufacturing, which is an emerging technology. More importantly, manufacturing constraints such as fiber cutting frequency, printing time and total turning angle of fibers are considered within the process planning for the first time.

Braess paradox in a quantum network

Dietrich Braess while working on traffic modelling, noticed that traffic flow in a network can be worsened by adding extra edges to an existing network. This seemingly counterintuitive phenomenon is known as the Braess paradox. We consider a quantum network, where edges represent shared entangled states between spatially separated parties(nodes). The goal is to entangle two previously uncorrelated nodes using entanglement swappings. The amount of entanglement between the distant nodes is quantified by the average concurrence of the states established, as a result of the entanglement swappings. We then introduce an additional edge of maximally entangled Bell states in the network. We show that the introduction of the additional maximally entangled states to this network leads to lower concurrence between the two previously un-correlated nodes. Thus we demonstrate the occurrence of a phenomenon in a quantum network that is analogous to the Braess' paradox in traffic networks.

An Exact Solution Approach for the Bus Line Planning Problem with Integrated Passenger Routing

The bus line planning problem or transit network design problem with integrated passenger routing is a challenging combinatorial problem. Although well-known benchmark instances for this problem have been available for decades, the state of the art lacks optimal solutions for these instances. The branch and bound algorithm, presented in this paper, introduces three novel concepts to determine these optimal solutions: (1) a new line pool generation method based on dominance, (2) the introduction of essential links, i.e., links which can be determined beforehand and must be present in the optimal solution, and (3) a new network representation based on adding only extra edges. Next to presenting the newly obtained optimal solutions, each of the abovementioned concepts is examined in isolation in the experiments, and it is shown that they contribute significantly to the success of the algorithm.

The number of triangles is more when they have no common vertex

By the theorem of Mantel (1907) it is known that a graph with n vertices and ⌊n24⌋+1 edges must contain a triangle. A theorem of Erdős gives a strengthening: there are not only one, but at least ⌊n2⌋ triangles. We give a further improvement: if there is no vertex contained by all triangles then there are at least n−2 of them. There are some natural generalizations when (a) complete graphs are considered (rather than triangles), (b) the graph has t extra edges (not only one) or (c) it is supposed that there are no s vertices such that every triangle contains one of them. We were not able to prove these generalizations, they are posed as conjectures.

Ramsey games near the critical threshold

A well‐known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if , then the random graph Gn, p is a.a.s. H‐Ramsey, that is, any 2‐coloring of its edges contains a monochromatic copy of H. Aside from a few simple exceptions, the corresponding 0‐statement also holds, that is, there exists c > 0 such that whenever the random graph Gn, p is a.a.s. not H‐Ramsey. We show that near this threshold, even when Gn, p is not H‐Ramsey, it is often extremely close to being H‐Ramsey. More precisely, we prove that for any constant c > 0 and any strictly 2‐balanced graph H, if , then the random graph Gn, p a.a.s. has the property that every 2‐edge‐coloring without monochromatic copies of H cannot be extended to an H‐free coloring after extra random edges are added. This generalizes a result by Friedgut, Kohayakawa, Rödl, Ruciński, and Tetali, who in 2002 proved the same statement for triangles, and addresses a question raised by those authors. We also extend a result of theirs on the three‐color case and show that these theorems need not hold when H is not strictly 2‐balanced.

A Novel Method for Scheduling of Wireless Ad Hoc Networks in Polynomial Time

In this article, we address the scheduling problem in wireless ad hoc networks by exploiting the computational advantage that comes when scheduling problems can be represented by claw-free conflict graphs where we consider a wireless broadcast medium. It is possible to formulate a scheduling problem of broadcast transmissions as finding the maximum weighted independent set (MWIS) in the conflict graph of the network. Finding the MWIS of a general graph is NP-hard leading to an NP-hard complexity of scheduling. In a claw-free conflict graph, MWIS may be found in polynomial time leading to a throughput-optimal scheduling. We show that the conflict graphs of certain wireless ad hoc networks are claw-free. In order to obtain claw-free conflict graphs in general networks, we suggest introducing additional conflicts (edges) with the aim of keeping the decrease in MWIS size minimal. To this end, we introduce an iterative optimization problem to decide where to introduce edges and investigate its efficient implementation. We conclude that the claw breaking method by adding extra edges can perform very close to optimal scenario and better than the polynomial time maximal independent set scheduling benchmark under the necessary assumptions.

Ramsey and Gallai–Ramsey numbers for stars with extra independent edges

Given a graph G and a positive integer k, define the Gallai–Ramsey number to be the minimum number of vertices n such that any k-edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this paper, we obtain general upper and lower bounds on the Gallai–Ramsey numbers for the graph G=Str obtained from a star of order t by including r extra independent edges between leaves of the star so there are r triangles and t−2r−1 pendant edges in Str. We also prove some sharp results when r=2.

Graph Sparsification, Spectral Sketches, and Faster Resistance Computation via Short Cycle Decompositions

We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition -- a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus few extra edges. A simple observation gives that every graph G on n vertices with m edges can be decomposed in $O(mn)$ time into cycles of length at most $2\log n$, and at most $2n$ extra edges. We give an $m^{1+o(1)}$ time algorithm for constructing a short cycle decomposition, with cycles of length $n^{o(1)}$, and $n^{1+o(1)}$ extra edges. These decompositions enable us to make progress on several open questions: * We give an algorithm to find $(1\pm\epsilon)$-approximations to effective resistances of all edges in time $m^{1+o(1)}\epsilon^{-1.5}$, improving over the previous best of $\tilde{O}(\min\{m\epsilon^{-2},n^2 \epsilon^{-1}\})$. This gives an algorithm to approximate the determinant of a Laplacian up to $(1\pm\epsilon)$ in $m^{1 + o(1)} + n^{15/8+o(1)}\epsilon^{-7/4}$ time. * We show existence and efficient algorithms for constructing graphical spectral sketches -- a distribution over sparse graphs H such that for a fixed vector $x$, we have w.h.p. $x'L_Hx=(1\pm\epsilon)x'L_Gx$ and $x'L_H^+x=(1\pm\epsilon)x'L_G^+x$. This implies the existence of resistance-sparsifiers with about $n\epsilon^{-1}$ edges that preserve the effective resistances between every pair of vertices up to $(1\pm\epsilon).$ * By combining short cycle decompositions with known tools in graph sparsification, we show the existence of nearly-linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians. Improved algorithms for constructing short cycle decompositions will lead to improvements for each of the above results.

Some exact values for p(t , d)

Let G = (V, E) be a graph. When a graph is used to model the linkage structure of communication networks, the diameter of a graph gives the length of longest path among all the shortest paths between any two vertices of the graph, F(t, d) denoted the minimum diameter of an altered graph obtained by adding t – extra edges to a graph with diameter d. Let P(t, d) denoted the minimum diameter of a graph obtained by adding t – extra edges to a path with diameter d. Knowing that F(t, d) = p(t, d). In this paper we prove that p(t, 3t) = 4, for t ≥ 4, p(t, d) = 4 where (t ≥ 4 and 3t +1 ≤ d ≤ 3t + 3) and show a construction to p(t, d) ≤ k + 3 where (3 ≤ t ≤ 5) and d = k(t + 1) + 8.

Continuous toolpath planning in a graphical framework for sparse infill additive manufacturing

We develop a framework that creates a new polygonal mesh representation of the sparse infill domain of a layer-by-layer 3D printing job. We guarantee the existence of a single, continuous tool path covering each connected piece of the domain in every layer in this graphical model. We also present a tool path algorithm that traverses each such continuous tool path with no crossovers.The key construction at the heart of our framework is a novel Euler transformation which converts a 2-dimensional cell complex K into a new 2-complex Kˆ such that every vertex in the 1-skeleton Gˆ of Kˆ has even degree. Hence Gˆ is Eulerian, and an Eulerian tour can be followed to print all edges in a continuous fashion without stops.We start with a mesh K of the union of polygons obtained by projecting all layers to the plane. First we compute its Euler transformation Kˆ. In the slicing step, we clip Kˆ at each layer using its polygon to obtain a complex that may not necessarily be Euler. We then patch this complex by adding edges such that any odd-degree nodes created by slicing are transformed to have even degrees again. We print extra support edges in place of any segments left out to ensure there are no edges without support in the next layer above. These support edges maintain the Euler nature of the complex. Finally, we describe a tree-based search algorithm that builds the continuous tool path by traversing “concentric” cycles in the Euler complex. Our algorithm produces a tool path that avoids material collisions and crossovers, and can be printed in a continuous fashion irrespective of complex geometry or topology of the domain (e.g., holes).We implement our test our framework on several 3D objects. Apart from standard geometric shapes including a nonconvex star, we demonstrate the framework on the Stanford bunny. Several intermediate layers in the bunny have multiple components as well as complicated geometries.

A comparison of neighbourhood relations based on ordinary Delaunay diagrams and area Delaunay diagrams: an application to define the neighbourhood relations of buildings

ABSTRACT The aim of this article is to describe a convenient but robust method for defining neighbourhood relations among buildings based on ordinary Delaunay diagrams (ODDs) and area Delaunay diagrams (ADDs). ODDs and ADDs are defined as a set of edges connecting the generators of adjacent ordinary Voronoi cells (points representing centroids of building polygons) and a set of edges connecting two centroids of building polygons, which are the generators of adjacent area Voronoi cells, respectively. Although ADDs are more robust than ODDs, computation time of ODDs is shorter than that of ADDs (the order of their computation time complexity is O(nlogn)). If ODDs can approximate ADDs with a certain degree of accuracy, the former can be used as an alternative. Therefore, we computed the ratio of the number of ADD edges to that of ODD edges overlapping ADDs at building and regional scales. The results indicate that: (1) for approximately 60% of all buildings, ODDs can exactly overlap ADDs with extra ODD edges; (2) at a regional scale, ODDs can overlap approximately 90% of ADDs with 10% extra ODD edges; and (3) focusing on judging errors, although ADDs are more accurate than ODDs, the difference is only approximately 1%.