On the VC-Dimension of Unique Round-Trip Shortest Path Systems

Area of Science:

• Graph Theory
• Computational Complexity
• Machine Learning Theory

Background:

• The VC-dimension is a key concept in learning theory, increasingly applied to graph algorithm analysis.
• Unique round-trip shortest path set systems (URTSP) are derived from vertex sets within unique round-trip shortest paths in directed graphs.

Purpose of the Study:

• To investigate and bound the VC-dimension of URTSP in directed graphs.
• To explore the implications of these bounds for graph algorithms, specifically the k-round-trip shortest path cover problem.

Main Methods:

• Theoretical analysis of set systems induced by shortest paths in directed graphs.
• Derivation of upper bounds for the VC-dimension of URTSP.
• Application of VC-dimension bounds to solve the minimum k-round-trip shortest path cover problem.

Main Results:

• The VC-dimension of URTSP is shown to be potentially greater than 3, unlike related shortest path set systems.
• A tight upper bound of 32 is established for the VC-dimension of URTSP.
• An upper bound is derived for the size of the vertex set in the k-round-trip shortest path cover problem.

Conclusions:

• The VC-dimension of URTSP is bounded, providing insights into the complexity of related graph problems.
• The findings contribute to the design and analysis of efficient graph algorithms.
• The derived bounds have practical implications for applications like facility placement.