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How Fast Can We Reach a Target Vertex in Stochastic Temporal Graphs?

Authors Eleni C. Akrida , George B. Mertzios , Sotiris Nikoletseas, Christoforos Raptopoulos , Paul G. Spirakis , Viktor Zamaraev



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Author Details

Eleni C. Akrida
  • Department of Computer Science, University of Liverpool, UK
George B. Mertzios
  • Department of Computer Science, Durham University, UK
Sotiris Nikoletseas
  • Computer Engineering & Informatics Department, University of Patras, and CTI, Greece
Christoforos Raptopoulos
  • Computer Engineering & Informatics Department, University of Patras, and CTI, Greece
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, UK
  • Computer Engineering & Informatics Department, University of Patras, Greece
Viktor Zamaraev
  • Department of Computer Science, Durham University, UK

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Eleni C. Akrida, George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, Paul G. Spirakis, and Viktor Zamaraev. How Fast Can We Reach a Target Vertex in Stochastic Temporal Graphs?. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 131:1-131:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.131

Abstract

Temporal graphs are used to abstractly model real-life networks that are inherently dynamic in nature, in the sense that the network structure undergoes discrete changes over time. Given a static underlying graph G=(V,E), a temporal graph on G is a sequence of snapshots {G_t=(V,E_t) subseteq G: t in N}, one for each time step t >= 1. In this paper we study stochastic temporal graphs, i.e. stochastic processes G={G_t subseteq G: t in N} whose random variables are the snapshots of a temporal graph on G. A natural feature of stochastic temporal graphs which can be observed in various real-life scenarios is a memory effect in the appearance probabilities of particular edges; that is, the probability an edge e in E appears at time step t depends on its appearance (or absence) at the previous k steps. In this paper we study the hierarchy of models memory-k, k >= 0, which address this memory effect in an edge-centric network evolution: every edge of G has its own probability distribution for its appearance over time, independently of all other edges. Clearly, for every k >= 1, memory-(k-1) is a special case of memory-k. However, in this paper we make a clear distinction between the values k=0 ("no memory") and k >= 1 ("some memory"), as in some cases these models exhibit a fundamentally different computational behavior for these values of k, as our results indicate. For every k >= 0 we investigate the computational complexity of two naturally related, but fundamentally different, temporal path (or journey) problems: {Minimum Arrival} and {Best Policy}. In the first problem we are looking for the expected arrival time of a foremost journey between two designated vertices {s},{y}. In the second one we are looking for the expected arrival time of the best policy for actually choosing a particular {s}-{y} journey. We present a detailed investigation of the computational landscape of both problems for the different values of memory k. Among other results we prove that, surprisingly, {Minimum Arrival} is strictly harder than {Best Policy}; in fact, for k=0, {Minimum Arrival} is #P-hard while {Best Policy} is solvable in O(n^2) time.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Temporal network
  • stochastic temporal graph
  • temporal path
  • #P-hard problem
  • polynomial-time approximation scheme

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