Parameterized temporal exploration problems

In this paper we study the fixed-parameter tractability of the problem of deciding whether a given temporal graph G admits a temporal walk that visits all vertices (temporal exploration) or, in some problem variants, a certain subset of the vertices. Formally, a temporal graph is a sequence G = hG1, . . . , GLi of graphs with V (Gt) = V (G) and E(Gt) ⊆ E(G) for all t ∈ [L] and some underlying graph G, and a temporal walk is a timerespecting sequence of edge-traversals. We consider both the strict variant, in which edges must be traversed in strictly increasing timesteps, and the non-strict variant, in which an arbitrary number of edges can be traversed in each timestep. For both variants, we give FPT algorithms for the problem of finding a temporal walk that visits a given set X of vertices, parameterized by |X|, and for the problem of finding a temporal walk that visits at least k distinct vertices in V (G), parameterized by k. We also show W[2]-hardness for a set version of the temporal exploration problem for both variants. For the non-strict variant, we give an FPT algorithm for the temporal exploration problem parameterized by the lifetime of the input graph, and we show that the temporal exploration problem can be solved in polynomial time if the graph in each timestep has at most two connected components

Temporal graph exploration: restrictions and relaxations

This thesis considers the problem of exploring temporal graphs. A temporal graph G = hG1; :::;GLi of order n is a sequence of L undirected graphs (or layers) indexed by the timesteps t 2 f1; : : : ;Lg, such that V (G1) = V (G) and  E(Gt)  E(G) for some underlying graph G with order n. To explore G is to visit each vertex at least once via a sequence of edge-traversals (called an exploration schedule), with each consecutive edge traversed during a timestep strictly greater than the last. The arrival time of an the timestep during which the last unvisited vertex is reached for the first time.There exists an algorithm producing exploration schedules with arrival time O(n2) for any always-connected (i.e., Gt is connected for all t 2 f1; : : : ;Lg) temporal graph, and an infinite family F of always-connected temporal graphs for which any exploration schedule has arrival time (n2) [38, 86]. We isolate a number of characteristics held by the members of F and prove lower/upper bounds on the arrival time of exploration schedules for temporal graphs that are restricted from possessing them. First, we consider structural restrictions in which an input temporal graph has (1) degree upper bounded by in each layer; and (2) at most k edges `missing' from the underlying graph in each layer; subquadratic upper bounds are proved in each case. We then consider `relaxed' exploration schedules that can traverse a ?nite number of edges ( 1) in each timestep, focusing on the cases when 2 or n=k traversals are allowed. We also consider, from a complexity standpoint, a number of relaxed problem variants, in which (1) less than n vertices are required to be explored by a candidate, and (2) an unlimited but ?nite number of edge traversals can be made by a candidate exploration schedule, providing both FPT-membership results and hardness/NP-completeness results.</p

Exploration of k-Edge-Deficient Temporal Graphs

A temporal graph with lifetime L is a sequence of L graphs G1, . . . , GL, called layers, all of which have the same vertex set V but can have different edge sets. The underlying graph is the graph with vertex set V that contains all the edges that appear in at least one layer. The temporal graph is always-connected if each layer is a connected graph, and it is k-edge-decient if each layer contains all except at most k edges of the underlying graph. For a given start vertex s, a temporal exploration is a temporal walk that starts at s, traverses at most one edge in each layer, and visits all vertices of the temporal graph. We show that always- connected, k-edge-decient temporal graphs with sucient lifetime can always be explored in O(kn log n) time steps. We also construct always-connected, k-edge- decient temporal graphs for which any exploration requires (n log k) time steps. For always-connected, 1-edge-decient temporal graphs, we show that O(n) time steps suce for temporal exploration

Faster Exploration of Degree-Bounded Temporal Graphs

A temporal graph can be viewed as a sequence of static graphs indexed by discrete time steps. The vertex set of each graph in the sequence remains the same; however, the edge sets are allowed to differ. A natural problem on temporal graphs is the Temporal Exploration problem (TEXP): given, as input, a temporal graph G of order n, we are tasked with computing an exploration schedule (i.e., a temporal walk that visits all vertices in G), such that the time step at which the walk arrives at the last unvisited vertex is minimised (we refer to this time step as the arrival time). It can be easily shown that general temporal graphs admit exploration schedules with arrival time no greater than O(n^2). Moreover, it has been shown previously that there exists an infinite family of temporal graphs for which any exploration schedule has arrival time Ω(n^2), making these bounds tight for general TEXP instances. We consider restricted instances of TEXP, in which the temporal graph given as input is, in every time step, of maximum degree d; we show an O(n^2/log n) bound on the arrival time when d is constant, and an O(d log d · n^2 / log n) bound when d is given as some function of n

Parameterized temporal exploration problems

In this paper we study the fixed-parameter tractability of the problem of deciding whether a given temporal graph G admits a temporal walk that visits all vertices (temporal exploration) or, in some problem variants, a certain subset of the vertices. Formally, a temporal graph is a sequence G = hG1, ..., GLi of graphs with V (Gt) = V (G) and E(Gt) ⊆ E(G) for all t ∈ [L] and some underlying graph G, and a temporal walk is a time-respecting sequence of edge-traversals. For the strict variant, in which edges must be traversed in strictly increasing timesteps, we give FPT algorithms for the problem of finding a temporal walk that visits a given set X of vertices, parameterized by |X|, and for the problem of finding a temporal walk that visits at least k distinct vertices in V , parameterized by k. For the non-strict variant, in which an arbitrary number of edges can be traversed in each timestep, we parameterize by the lifetime L of the input graph and provide an FPT algorithm for the temporal exploration problem. We also give additional FPT or W[2]-hardness results for further problem variants

Two Moves per Time Step Make a Difference

A temporal graph is a graph whose edge set can change over time. We only require that the edge set in each time step forms a connected graph. The temporal exploration problem asks for a temporal walk that starts at a given vertex, moves over at most one edge in each time step, visits all vertices, and reaches the last unvisited vertex as early as possible. We show in this paper that every temporal graph with n vertices can be explored in O(n^{1.75}) time steps provided that either the degree of the graph is bounded in each step or the temporal walk is allowed to make two moves per step. This result is interesting because it breaks the lower bound of Omega(n^2) steps that holds for the worst-case exploration time if only one move per time step is allowed and the graph in each step can have arbitrary degree. We complement this main result by a logarithmic inapproximability result and a proof that for sparse temporal graphs (i.e., temporal graphs with O(n) edges in the underlying graph) making O(1) moves per time step can improve the worst-case exploration time at most by a constant factor