Finding Path Motifs in Large Temporal Graphs Using Algebraic Fingerprints

Finding Colorful Paths in Temporal Graphs

“…Finding paths is a basic problem in graph theory [7] and several variants have been studied, including finding a shortest path between two vertices and finding a longest path in a graph. Recently, these problems have been considered for real-world data that need a description of the vertex properties and dynamics of the relations [15]. For these data, a richer representation with respect to the classical graph model has to be introduced, for example by associating labels or colors with vertices and by representing the evolution of relations with a temporal graph.…”

“…A temporal path in a temporal graph is a path in which the timestamps of consecutive edges are strictly increasing, thus representing a path that does not violate the time constraint specified by the timestamps of the edges. The problem we consider is a variant of the one considered in [15], that asks for a temporal path that exactly matches a multiset of colors (called motif in [15]). As outlined in [15], this problem has several applications, for example in tour recommendations [5,9], where vertices correspond to interesting locations, colors represent activities available in locations, edges correspond to transportation links between different locations (a timestamp is associated for example to departure time).…”

“…The problem we consider is a variant of the one considered in [15], that asks for a temporal path that exactly matches a multiset of colors (called motif in [15]). As outlined in [15], this problem has several applications, for example in tour recommendations [5,9], where vertices correspond to interesting locations, colors represent activities available in locations, edges correspond to transportation links between different locations (a timestamp is associated for example to departure time). A set (or a multiset) of colors represents activities a tourist may be interested into and a path associated with different colors is then a suggestion of the activities that can be carried out respecting the time constraints.…”

“…Several variants of the problem of finding a temporal path in a temporal vertex-colored graph that matches a given multiset of colors (called motif) have been introduced in [15]. It is shown in [15] that these variants of the problem are NP-complete, but fixed-parameter tractable when parameterized by the size of the motif.…”

“…In Section 4 we present a heuristic for Max CPTG, as our aim is to design a method that is applicable even for a large number of colors. Notice that the methods proposed in [15] are for different variants of the problem, where all the colors of the motif have to be included in a solution. Moreover, the methods proposed in [15] are fixed-parameter algorithms, where the parameter is the size of the motif, hence the running time of these latter algorithms is exponential in the size of the motif, leading to methods that are able to process motifs of moderate size (up to 18 colors are considered in [15]).…”