scholarly journals Interference-free Walks in Time: Temporally Disjoint Paths

2021 ◽  
Author(s):  
Nina Klobas ◽  
George B. Mertzios ◽  
Hendrik Molter ◽  
Rolf Niedermeier ◽  
Philipp Zschoche
Keyword(s):  
Disjoint Paths ◽  
Constant Number ◽  
Underlying Graph ◽  
Traffic Routing ◽  
Temporal Graph ◽  
The One ◽  
Temporal Graphs ◽  
General Graphs ◽  
Temporal Path ◽  
Np Hardness

We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The "walk version" is W[1]-hard when parameterized by the number of routes. However, it is polynomial-time solvable for any constant number of walks. The "path version" remains NP-hard even if we want to find only two temporally disjoint paths. On the other extreme, restricting the input temporal graph to have a path as underlying graph, quite counterintuitively, we find NP-hardness in general but also identify natural tractable cases.

Time-topology analysis

2021 ◽  
Vol 14 (13) ◽  
pp. 3322-3334
Author(s):  
Yunkai Lou ◽  
Chaokun Wang ◽  
Tiankai Gu ◽  
Hao Feng ◽  
Jun Chen ◽  
...  
Keyword(s):  
Temporal Information ◽  
Time Dimension ◽  
Topology Analysis ◽  
Topological Information ◽  
Analysis Methods ◽  
Short Period ◽  
Pruning Strategy ◽  
Temporal Graph ◽  
Types Of Information ◽  
Temporal Graphs

Many real-world networks have been evolving, and are finely modeled as temporal graphs from the viewpoint of the graph theory. A temporal graph is informative, and always contains two types of information, i.e., the temporal information and topological information, where the temporal information reflects the time when the relationships are established, and the topological information focuses on the structure of the graph. In this paper, we perform time-topology analysis on temporal graphs to extract useful information. Firstly, a new metric named T-cohesiveness is proposed to evaluate the cohesiveness of a temporal subgraph. It defines the cohesiveness of a temporal subgraph from the time and topology dimensions jointly. Specifically, given a temporal graph G s = ( Vs , ε Es ), cohesiveness in the time dimension reflects whether the connections in G s happen in a short period of time, while cohesiveness in the topology dimension indicates whether the vertices in V s are densely connected and have few connections with vertices out of G s . Then, T-cohesiveness is utilized to perform time-topology analysis on temporal graphs, and two time-topology analysis methods are proposed. In detail, T-cohesiveness evolution tracking traces the evolution of the T-cohesiveness of a subgraph, and combo searching finds out all the subgraphs that contain the query vertex and have T-cohesiveness larger than a given threshold. Moreover, a pruning strategy is proposed to improve the efficiency of combo searching. Experimental results confirm the efficiency of the proposed time-topology analysis methods and the pruning strategy.

scholarly journals Approximating the Temporal Neighbourhood Function of Large Temporal Graphs

Algorithms ◽  
2019 ◽  
Vol 12 (10) ◽  
pp. 211 ◽  
Author(s):  
Pierluigi Crescenzi ◽  
Clémence Magnien ◽  
Andrea Marino
Keyword(s):  
Public Transportation ◽  
Arrival Time ◽  
Time Interval ◽  
Temporal Network ◽  
Temporal Networks ◽  
Starting Time ◽  
Neighborhood Function ◽  
Temporal Graphs ◽  
Temporal Path

Temporal networks are graphs in which edges have temporal labels, specifying their starting times and their traversal times. Several notions of distances between two nodes in a temporal network can be analyzed, by referring, for example, to the earliest arrival time or to the latest starting time of a temporal path connecting the two nodes. In this paper, we mostly refer to the notion of temporal reachability by using the earliest arrival time. In particular, we first show how the sketch approach, which has already been used in the case of classical graphs, can be applied to the case of temporal networks in order to approximately compute the sizes of the temporal cones of a temporal network. By making use of this approach, we subsequently show how we can approximate the temporal neighborhood function (that is, the number of pairs of nodes reachable from one another in a given time interval) of large temporal networks in a few seconds. Finally, we apply our algorithm in order to analyze and compare the behavior of 25 public transportation temporal networks. Our results can be easily adapted to the case in which we want to refer to the notion of distance based on the latest starting time.

Finding multiple induced disjoint paths in general graphs

2011 ◽  
Vol 111 (20) ◽  
pp. 1022-1026
Author(s):  
Kejia Zhang ◽  
Hong Gao ◽  
Jianzhong Li
Keyword(s):  
Disjoint Paths ◽  
General Graphs

scholarly journals Two Influence Maximization Games on Graphs Made Temporal

2021 ◽  
Author(s):  
Niclas Boehmer ◽  
Vincent Froese ◽  
Julia Henkel ◽  
Yvonne Lasars ◽  
Rolf Niedermeier ◽  
...  
Keyword(s):  
Nash Equilibria ◽  
Dynamic Nature ◽  
Games On Graphs ◽  
Graph Games ◽  
Graph Classes ◽  
Temporal Graph ◽  
Influence Spreading ◽  
Temporal Graphs ◽  
Competitive Diffusion ◽  
Over Time

To address the dynamic nature of real-world networks, we generalize competitive diffusion games and Voronoi games from static to temporal graphs, where edges may appear or disappear over time. This establishes a new direction of studies in the area of graph games, motivated by applications such as influence spreading. As a first step, we investigate the existence of Nash equilibria in competitive diffusion and Voronoi games on different temporal graph classes. Even when restricting our studies to temporal paths and cycles, this turns out to be a challenging undertaking, revealing significant differences between the two games in the temporal setting. Notably, both games are equivalent on static paths and cycles. Our two main technical results are (algorithmic) proofs for the existence of Nash equilibria in temporal competitive diffusion and temporal Voronoi games when the edges are restricted not to disappear over time.

PACKING A TRUCK — NOW WITH A TWIST!

2007 ◽  
Vol 17 (05) ◽  
pp. 505-527
Author(s):  
FRIEDRICH EISENBRAND ◽  
STEFAN FUNKE ◽  
ANDREAS KARRENBAUER ◽  
JOACHIM REICHEL ◽  
ELMAR SCHÖMER
Keyword(s):  
Early Stage ◽  
Practical Importance ◽  
Continuous Model ◽  
Optimization Techniques ◽  
New Approach ◽  
The Past ◽  
European Industry ◽  
Closing The Gap ◽  
The One ◽  
Np Hardness

In an industry project with a German car manufacturer we are faced with the challenge of placing a maximum number of uniform rigid rectangular boxes in the interior of a car trunk. The problem is of practical importance due to a European industry norm which requires car manufacturers to state the trunk volume according to this measure. No really satisfactory automated solution for this problem has been known in the past. In spite of its NP hardness, combinatorial optimization techniques, which consider only grid-aligned placements, produce solutions which are very close to the one achievable by a human expert in several hours of tedious work. The remaining gap is mostly due to the constraints imposed by the chosen grid. In this paper we present a new approach which combines the grid-based combinatorial method with Simulated Annealing on a continuous model. This allows us to explore arbitrary orientations and placements of boxes, hence closing the gap even further, and – in some cases – even surpass the manual expert solution. The implemented software system allows our industrial partner to incorporate the trunk volume in a very early stage of the car design process without relying on a repeated and cumbersome manual evaluation of the volume.

One-to-One Disjoint Path Covers on WK-Networks

2014 ◽  
Vol 651-653 ◽  
pp. 1875-1881
Author(s):  
Lan Tao You ◽  
Yue Juan Han
Keyword(s):  
Complete Graph ◽  
Disjoint Paths ◽  
Disjoint Path ◽  
One To One ◽  
The One

The WK-recursive network has received much attention due to its many attractive properties. In this paper, we consider the one-to-one disjoint path covers properties of the WK-recursive network. We use K(d, t) to denote the WK-recursive network of level t, each of which basic modules is a d-vertex complete graph, where d > 1 and t ≥ 1. We prove that for any two distinct vertices u and v, there exist d-1 node-disjoint paths whose union covers all vertices of K(d, t) for d ≥ 3 and t ≥ 1. The results is optimal for vertices in different Kj(d, t − 1) for t ≥ 2, since each Kj(d, t − 1) with 1 ≤ j ≤ d has d − 1 open edges.

scholarly journals On the maximum avoidance of inbreeding

1963 ◽  
Vol 4 (3) ◽  
pp. 399-415 ◽  
Author(s):  
Motoo Kimura ◽  
James F. Crow
Keyword(s):  
Gene Frequency ◽  
Mating Systems ◽  
Initial Rate ◽  
Constant Number ◽  
Random Drift ◽  
Pair Mating ◽  
Related Individuals ◽  
The One ◽  
Image Position ◽  
The Right

Mating systems in which the least related individuals are mated have been designated by Wright as having maximum avoidance of inbreeding. For such systems the initial rate of decrease in heterozygosity is minimum. However, some other systems have a lower rate of decrease in later generations.Circular mating, in which each individual is mated with the one to his right and to his left, leads to an asymptotic rate of decrease in heterozygosity of 1– λ ˜ π2/(2N + 4)2 compared with 1/4N for maximum avoidance systems. Circular pair mating, in which for example each male progeny is moved one cage to the right, leads to 1– λ ~ π2/(N + 12)2. Other similar systems are discussed.For minimum gene frequency drift, a mating system should have a constant number of progeny per parent and the population should be broken up as rapidly as possible into the maximum number of lines. The gene frequency variance at generation T within a line iswhere N is the number in the line and Ht is the proportion of heterozygotes in generation t. Although the three mating systems, circular, circular pair, and maximum avoidance (and many others) have the same amount of random drift ultimately, at any generation circular mating has the smallest drift variance, VT, and circular pair next smallest.

scholarly journals On the Entanglement Cost of One-Shot Compression

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 286
Author(s):  
Shima Bab Hadiashar ◽  
Ashwin Nayak
Keyword(s):  
Boolean Functions ◽  
Communication Complexity ◽  
Constant Error ◽  
Communication Cost ◽  
Constant Number ◽  
Information Cost ◽  
State Splitting ◽  
The One ◽  
The Cost ◽  
The Given

We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.

scholarly journals Finding Minimum-Weight Link-Disjoint Paths with a Few Common Nodes

2020 ◽  
Vol 34 (01) ◽  
pp. 938-945
Author(s):  
Binglin Tao ◽  
Mingyu Xiao ◽  
Jingyang Zhao
Keyword(s):  
Network Optimization ◽  
Minimum Weight ◽  
Network Survivability ◽  
Disjoint Paths ◽  
Network Resources ◽  
Restricted Version ◽  
Backup Path ◽  
Link Weight ◽  
Np Hardness ◽  
Full Protection

Network survivability has drawn certain interest in network optimization. However, the demand for full protection of a network is usually too restrictive. To overcome the limitation of geographical environments and to save network resources, we turn to establish backup networks allowing a few common nodes. It comes out the problem of finding k link-disjoint paths between a given pair of source and sink in a network such that the number of common nodes shared by at least two paths is bounded by a constant and the total link weight of all paths is minimized under the above constraints. For the case k = 2, where we have only one backup path, several fast algorithms have been developed in the literature. For the case k > 2, little results are known. In this paper, we first establish the NP-hardness of the problem with general k. Motivated by the situation that each node in a network may have a capability of multicasting, we also study a restricted version with one more requirement that each node can be shared by at most two paths. For the restricted version, we build an ILP model and design a fast algorithm by using the techniques of augmenting paths and splitting nodes. Furthermore, experimental results on synthetic and real networks show that our algorithm is effective in practice.

scholarly journals Distributed temporal graph analytics with GRADOOP

2021 ◽  
Author(s):  
Christopher Rost ◽  
Kevin Gomez ◽  
Matthias Täschner ◽  
Philip Fritzsche ◽  
Lucas Schons ◽  
...  
Keyword(s):  
Graph Model ◽  
Lessons Learned ◽  
Temporal Property ◽  
Graph Analytics ◽  
Matching Graph ◽  
Temporal Graph ◽  
Scalable Analysis ◽  
Temporal Graphs ◽  
Distributed Analytics ◽  
Difference Pattern

AbstractTemporal property graphs are graphs whose structure and properties change over time. Temporal graph datasets tend to be large due to stored historical information, asking for scalable analysis capabilities. We give a complete overview of Gradoop, a graph dataflow system for scalable, distributed analytics of temporal property graphs which has been continuously developed since 2005. Its graph model TPGM allows bitemporal modeling not only of vertices and edges but also of graph collections. A declarative analytical language called GrALa allows analysts to flexibly define analytical graph workflows by composing different operators that support temporal graph analysis. Built on a distributed dataflow system, large temporal graphs can be processed on a shared-nothing cluster. We present the system architecture of Gradoop, its data model TPGM with composable temporal graph operators, like snapshot, difference, pattern matching, graph grouping and several implementation details. We evaluate the performance and scalability of selected operators and a composed workflow for synthetic and real-world temporal graphs with up to 283 M vertices and 1.8 B edges, and a graph lifetime of about 8 years with up to 20 M new edges per year. We also reflect on lessons learned from the Gradoop effort.

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