[Paper Review] DMVP: Foremost Waypoint Coverage of Time-Varying Graphs

Eric Aaron, Danny Kriz̧anc+1 more|arXiv (Cornell University)|Jul 27, 2014

Optimization and Search Problems22 references4 citations

TL;DR

This paper introduces the Dynamic Map Visitation Problem (DMVP) on time-varying graphs (TVGs), analyzing its computational complexity across three classes of dynamics: R (recurring edges), B (bounded reappearances), and P (periodic edges). It establishes severe inapproximability in R, limited approximability in B, and tractability in P for specific topologies, with fixed-parameter tractability identified in R for certain graph structures.

ABSTRACT

We consider the Dynamic Map Visitation Problem (DMVP), in which a team of agents must visit a collection of critical locations as quickly as possible, in an environment that may change rapidly and unpredictably during the agents' navigation. We apply recent formulations of time-varying graphs (TVGs) to DMVP, shedding new light on the computational hierarchy $\mathcal{R} \supset \mathcal{B} \supset \mathcal{P}$ of TVG classes by analyzing them in the context of graph navigation. We provide hardness results for all three classes, and for several restricted topologies, we show a separation between the classes by showing severe inapproximability in $\mathcal{R}$, limited approximability in $\mathcal{B}$, and tractability in $\mathcal{P}$. We also give topologies in which DMVP in $\mathcal{R}$ is fixed parameter tractable, which may serve as a first step toward fully characterizing the features that make DMVP difficult.

Motivation & Objective

• To formalize the Dynamic Map Visitation Problem (DMVP) as a navigation challenge in time-varying graphs (TVGs), where edge availability changes dynamically.
• To analyze the computational complexity of DMVP across three TVG classes: R (recurring edges), B (bounded reappearances), and P (periodic edges).
• To identify topological and dynamic constraints that differentiate tractable from intractable instances of DMVP.
• To establish separations between the computational hardness of DMVP in R, B, and P classes using restricted graph topologies.
• To explore fixed-parameter tractability in R for specific graph families, laying groundwork for characterizing structural features that influence DMVP difficulty.

Proposed method

• Models the environment as a time-varying graph (TVG) with discrete time, undirected edges, and uniform traversal cost of 1.
• Defines the DMVP as a single-agent problem to visit all critical waypoints as quickly as possible under dynamic edge availability.
• Classifies TVGs into three classes: R (edges reappear eventually), B (edges reappear within a bounded time), and P (edges appear periodically).
• Applies temporal subgraph analysis to restrict the lifetime of the TVG and evaluate visitation feasibility within a time bound.
• Uses perfect bipartite matching algorithms to determine waiting-free traversals in star-like graphs (spiders) with uniform arm lengths.
• Employs dynamic programming and pattern-piecemealing techniques for optimal traversal sequences in periodic settings (P class).

Experimental results

Research questions

• RQ1 What is the computational complexity of DMVP in the R, B, and P classes of time-varying graphs?
• RQ2 Can there be a strict separation in complexity between R, B, and P classes for DMVP on restricted topologies?
• RQ3 Are there specific graph topologies for which DMVP in the R class is fixed-parameter tractable?
• RQ4 Is DMVP NP-hard in R but not in B for any class of underlying graphs, particularly stars with bounded degree?
• RQ5 Can optimal traversal sequences be efficiently computed for periodic edge dynamics (P class) over general trees?

Key findings

• DMVP is severely inapproximable in the R class, even for restricted topologies, indicating high computational hardness.
• In the B class, DMVP admits limited approximability, suggesting some algorithmic feasibility under bounded reappearances.
• For the P class, DMVP is tractable on trees, with a linear-time algorithm for p=2 agents and periodic dynamics.
• DMVP is solvable in polynomial time for spiders (stars with uniform arm lengths) when waiting is disallowed and time is tight.
• A waiting-free traversal in a spider with uniform arms reduces to a perfect bipartite matching problem, solvable in polynomial time.
• The paper conjectures that DMVP in R is tractable over graphs with polynomially many spanning trees, each with O(log n) leaves, and that this class may also characterize B.