[Paper Review] Finding Temporal Paths Under Waiting Time Constraints

Arnaud Casteigts, Anne-Sophie Himmel+2 more|arXiv (Cornell University)|Sep 13, 2019

Data Management and Algorithms69 references21 citations

TL;DR

This paper introduces the Restless Temporal Path problem, where temporal paths must respect a maximum waiting time Δ at each vertex, modeling scenarios like disease spread with immunity or delay-tolerant networking. The authors prove the problem is NP-hard and W[1]-hard even under strict constraints, but present FPT algorithms for parameters including path length, feedback edge number, and a novel timed feedback vertex number, offering tractability in structured settings.

ABSTRACT

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration Δ, referred to as Δ-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the "restless variant" of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the feedback vertex number or the pathwidth of the underlying graph. The main question thus is whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

Motivation & Objective

• To formalize and analyze the computational complexity of finding temporal paths under waiting time constraints, where each vertex can be visited at most once and waiting time is bounded by Δ.
• To model real-world processes such as disease transmission with immunity and packet routing in delay-tolerant networks using temporal graphs with bounded waiting times.
• To identify structural parameters that allow for fixed-parameter tractable (FPT) algorithms despite the problem's inherent hardness.
• To introduce and utilize a new parameter, the timed feedback vertex number, to capture fine-grained temporal structure for improved algorithmic efficiency.

Proposed method

• Define the Restless Temporal Path problem as finding a time-respecting path in a temporal graph where waiting time at each vertex is at most Δ.
• Prove NP-hardness and W[1]-hardness under the distance-to-disjoint-paths parameter, showing the problem is computationally hard even in restricted cases.
• Introduce the timed feedback vertex number, a novel parameter that combines temporal and structural graph features to enable FPT algorithms.
• Design FPT algorithms for three parameter classes: (1) output size (maximum path length), (2) classical graph parameters (e.g., feedback edge number), and (3) the new timed feedback vertex number.
• Use reduction rules and vertex duplication techniques to transform instances into equivalent ones with undeletable vertices, enabling application of known Subset Feedback Vertex Set algorithms.
• Leverage known FPT algorithms for Subset Feedback Vertex Set (e.g., Iwata et al.) to solve the transformed instances efficiently.

Experimental results

Research questions

• RQ1 Is the Restless Temporal Path problem fixed-parameter tractable when parameterized by the maximum path length?
• RQ2 Can the problem be solved efficiently when parameterized by classical graph parameters such as feedback edge number or pathwidth?
• RQ3 Does the introduction of a new temporal-structural parameter—timed feedback vertex number—enable FPT algorithms for this problem?
• RQ4 How does the problem's complexity change when the lifetime of the temporal graph is restricted to only three time steps?
• RQ5 Is the problem W[1]-hard when parameterized by the distance to disjoint paths in the underlying graph?

Key findings

• The Restless Temporal Path problem is NP-hard even when the temporal graph has only three time steps, indicating strong intractability in restricted settings.
• The problem is W[1]-hard when parameterized by the distance to disjoint paths of the underlying graph, which implies W[1]-hardness for feedback vertex number and pathwidth.
• An FPT algorithm exists for the problem when parameterized by the maximum path length, with running time linear in the graph size.
• An FPT algorithm is presented for the feedback edge number of the underlying graph, providing tractability under this structural parameter.
• The novel timed feedback vertex number parameter enables an FPT algorithm, and the problem can be solved in O(k²(|V| + |E|)) time using reductions to Subset Feedback Vertex Set.
• The problem remains hard even under strong constraints, but tractable for specific structural parameters, showing a clear dichotomy between hard and tractable cases.