The Complexity of Finding Small Separators in Temporal Graphs

Temporal graphs are graphs with time-stamped edges. We study the problem of finding a small vertex set (the separator) with respect to two designated terminal vertices such that the removal of the set eliminates all temporal paths connecting one terminal to the other. Herein, we consider two models of temporal paths: paths that pass through arbitrarily many edges per time step (non-strict) and paths that pass through at most one edge per time step (strict). Regarding the number of time steps of a temporal graph, we show a complexity dichotomy (NP-hardness versus polynomial-time solvability) for both problem variants. Moreover we prove both problem variants to be NP-complete even on temporal graphs whose underlying graph is planar. We further show that, on temporal graphs with planar underlying graph, if additionally the number of time steps is constant, then the problem variant for strict paths is solvable in quasi-linear time. Finally, we introduce and motivate the notion of a temporal core (vertices whose incident edges change over time). We prove that the non-strict variant is fixed-parameter tractable when parameterized by the size of the temporal core, while the strict variant remains NP-complete, even for constant-size temporal cores

On Exploring Temporal Graphs of Small Pathwidth

We show that the Temporal Graph Exploration Problem is NP-complete, even when the underlying graph has pathwidth 2 and at each time step, the current graph is connected

Enumeration of s-d separators in DAGs with application to reliability analysis in temporal graphs

Temporal graphs are graphs in which arcs have temporal labels, specifying at which time they can be traversed. Motivated by recent results concerning the reliability analysis of a temporal graph through the enumeration of minimal cutsets in the corresponding line graph, in this paper we attack the problem of enumerating minimal s-d separators in s-d directed acyclic graphs (in short, s-d DAGs), also known as 2-terminal DAGs or s-t digraphs. Our main result is an algorithm for enumerating all the minimal s-d separators in a DAG with O(nm) delay, where n and m are respectively the number of nodes and arcs, and the delay is the time between the output of two consecutive solutions. To this aim, we give a characterization of the minimal s-d separators in a DAG through vertex cuts of an expanded version of the DAG itself. As a consequence of our main result, we provide an algorithm for enumerating all the minimal s-d cutsets in a temporal graph with delay O(m3), where m is the number of temporal arcs

Robust Group Linkage

We study the problem of group linkage: linking records that refer to entities in the same group. Applications for group linkage include finding businesses in the same chain, finding conference attendees from the same affiliation, finding players from the same team, etc. Group linkage faces challenges not present for traditional record linkage. First, although different members in the same group can share some similar global values of an attribute, they represent different entities so can also have distinct local values for the same or different attributes, requiring a high tolerance for value diversity. Second, groups can be huge (with tens of thousands of records), requiring high scalability even after using good blocking strategies. We present a two-stage algorithm: the first stage identifies cores containing records that are very likely to belong to the same group, while being robust to possible erroneous values; the second stage collects strong evidence from the cores and leverages it for merging more records into the same group, while being tolerant to differences in local values of an attribute. Experimental results show the high effectiveness and efficiency of our algorithm on various real-world data sets