Counting Temporal Paths

TL;DR

This paper investigates the computational complexity of counting optimal temporal paths in dynamic graphs, establishing hardness results and developing specialized algorithms for certain cases.

Contribution

It systematically studies the parameterized and approximation complexity of #Temporal Path, providing new insights and algorithms for specific scenarios.

Findings

#Temporal Path is #P-hard in general.

Hard to approximate in the general case.

Developed exact and approximate FPT-algorithms for special cases.

Abstract

The betweenness centrality of a vertex v is an important centrality measure that quantifies how many optimal paths between pairs of other vertices visit v. Computing betweenness centrality in a temporal graph, in which the edge set may change over discrete timesteps, requires us to count temporal paths that are optimal with respect to some criterion. For several natural notions of optimality, including foremost or fastest temporal paths, this counting problem reduces to #Temporal Path, the problem of counting all temporal paths between a fixed pair of vertices; like the problems of counting foremost and fastest temporal paths, #Temporal Path is #P-hard in general. Motivated by the many applications of this intractable problem, we initiate a systematic study of the prameterised and approximation complexity of #Temporal Path. We show that the problem presumably does not admit an FPT-algorithm for the feedback vertex number of the static underlying graph, and that it is hard to approximate in general. On the positive side, we proved several exact and approximate FPT-algorithms for special cases.