Skyline-First Traversal as a Control Mechanism for Multi-Criteria Graph Search
Summary: arXiv:2604.19807v1 Announce Type: new
Abstract
In multi-criteria graph traversal, paths are compared via Pareto dominance, an ordering that identifies which paths are non-dominated, but says nothing about which path to expand next or when the search may stop. As a result, existing approaches rely on external mechanisms—heuristics, scalarization, or population-based exploration—while Pareto dominance remains confined to passive roles such as pruning or ranking.
This paper shows that, under constrained cost models, finite cost grids, Markovian transitions, and a nonzero progress measure, Pareto geometry alone is sufficient to drive both scheduling and termination. We show that extracting exclusively from the first Pareto layer, the skyline, induces a deterministic descent in a discrete completion potential, ensuring monotone progress toward solution completion.
Key Findings
- The extraction of paths from the first Pareto layer, or skyline, leads to a structured approach for graph traversal.
- A vector lower-bound certificate is introduced to provide a stopping condition that ensures dominance coverage of all remaining traversals without necessitating a predefined number of solutions.
- The framework eliminates the need for scalarization, heuristic guidance, or probabilistic models, repositioning Pareto dominance as an active driver rather than a passive filter.
Methodology
Our analysis establishes several key concepts:
- Deterministic Potential Descent: The framework guarantees a steady decline in the completion potential as the search progresses through the skyline paths.
- Certified Termination: The stopping condition based on dominance coverage ensures that the search does not prematurely conclude, thus covering all potential traversals.
- Uniform Bound on Layer Width: The geometry of the cost grid imposes a consistent width across Pareto layers, which aids in maintaining search efficiency.
- Greedy Cost-Space Dispersion: The approach promotes an efficient dispersion of search efforts within the skyline, enhancing the overall exploration of the graph.
Implications
The implications of this research are significant for future developments in multi-criteria graph search algorithms. By leveraging Pareto geometry, the proposed method provides a more efficient and deterministic framework that can adapt to various constrained environments. This positions the skyline-first approach as a viable alternative to traditional multi-criteria search methods that rely heavily on external strategies.
Conclusion
In conclusion, the skyline-first traversal as a control mechanism for multi-criteria graph search presents a groundbreaking shift in how paths are evaluated and progressed. This research not only offers a robust theoretical foundation but also opens avenues for practical applications in fields such as optimization, operations research, and artificial intelligence.