Real-weighted Diameter and Eccentricity of Minor-free and Bounded VC-dimension Graphs in Truly Subquadratic Time

Abstract: We present the first truly subquadratic time algorithm to compute diameter and eccentricity in real-weighted directed graphs with constant distance VC-dimension and strongly sublinear-sized balanced separators. This runs in $O(n<sup>{2-1/(2h-2)}\textrm{polylog}(n))$ time for real-weighted $K_h$-minor-free digraphs. Prior to this work, truly subquadratic time computation of diameter was only known for real-weighted planar graphs, while extensions to broader classes like minor-free graphs were restricted to unweighted settings. In particular, existing algorithms that use VC-dimension [Ducoffe, Habib, Viennot; SICOMP 2022][Le, Wulff-Nilsen; SODA 2024][Chan, Chang, Gao, Le, Kisfaludi-Bak, Zheng; FOCS 2025] work with small integer weights, but do not naturally generalize to real weights. We overcome this barrier by introducing a randomized search-to-decision reduction, demonstrating that VC-dimension is a sufficiently powerful tool in the real-weighted regime.

• The paper establishes the first truly subquadratic algorithm for computing exact eccentricities and diameter in real-weighted K_h-minor-free digraphs.
• It introduces a randomized search-to-decision reduction that leverages bounded VC-dimension to efficiently handle continuous real weights.
• The algorithm employs r-divisions, balanced separators, and low-crossing paths to achieve a runtime of O(n^(2-1/(2h-2)) · polylog(n)).

Real-weighted Diameter and Eccentricity in Minor-free and Bounded VC-dimension Graphs

Problem Context and Prior Work

The computation of diameter and eccentricities in sparse real-weighted directed graphs is central to algorithmic graph theory and diverse applications in network analysis and computational geometry. For general sparse graphs, the all-pairs shortest path (APSP) problem using Dijkstra's algorithm from every vertex yields an $O(n^2)$ time solution, where $n$ is the number of vertices. However, under SETH, exact quadratic-time lower bounds hold for general graphs, even unweighted and undirected, motivating algorithmic study in structured graphs, such as planar, bounded-treewidth, and minor-free graphs.

For planar graphs, subquadratic-time deterministic algorithms exist for both unweighted and real-weighted cases, using techniques such as abstract Voronoi diagrams and $r$-divisions, e.g., $O(n^{5/3})$ for diameter computation [Gawrychowski et al.]. Extension to minor-free graphs has been substantially more challenging, especially for the real-weighted and directed case. Prior to this work, truly subquadratic algorithms for diameter and eccentricity in minor-free graphs were limited to unweighted settings or required small integer weights. Attempts to generalize bounded VC-dimension techniques to real weights encountered significant obstacles due to the necessity for search-to-decision reductions that can handle the continuum of possible distances.

Main Contributions

This paper establishes the first truly subquadratic-time algorithm for computing exact eccentricities and diameter in real-weighted $K_h$-minor-free digraphs and, more generally, in directed graphs with constant distance VC-dimension and strongly sublinear-sized balanced separators. The algorithm runs in $O(n^{2-1/(2h-2)} \cdot \mathrm{polylog}(n))$ time for $K_h$-minor-free digraphs, where $h$ is the size of the forbidden minor.

The core technical innovation is a randomized search-to-decision reduction that uses random witnesses, allowing the extension of VC-dimension-based methods—which were previously limited to integer-weighted graphs—to the real-weighted setting. This demonstrates that VC-dimension can be leveraged as an algorithmic tool beyond the setting of combinatorial path counts, accepting arbitrary positive real edge weights.

The result generalizes to all monotone graph classes with polynomial expansion (i.e., those with strongly sublinear-sized balanced separators), provided the generalized distance VC-dimension is bounded. For such graph classes $G_\beta$ (with balanced separators of size $O(n^\beta)$ and distance VC-dimension $n$0), the algorithm computes all vertex eccentricities in $n$1 time.

Algorithmic Design and Analysis

The algorithm uses several structural decomposition and combinatorial techniques:

• $n$2-divisions and Balanced Separators: The graph is decomposed into pieces of size at most $n$3 with total boundary size $n$4. In $n$5-minor-free graphs, $n$6-divisions with total boundary $n$7 can be computed efficiently.
• VC-dimension Set Systems: The generalized distance VC-dimension of the graph enables the use of low-average crossing spanning paths, allowing efficient interval representations and sampling for the set of reachable vertices at a given distance.
• Random Witness Search-to-Decision Reduction: Instead of classic binary search on distances, which is infeasible for real weights due to the uncountable number of possible values, the algorithm employs a randomized process. Each time a guessed eccentricity is too small, a random witness outside the ball is found, which probabilistically halves the set of possible vertices at every round. With high probability, the number of rounds is $n$8 for all vertices simultaneously.
• Interval Merging and Sampling: Efficient union and interval representations are used for merging and sampling witness candidates. These operations are optimized using the properties of low crossing spanning paths derived from bounded VC-dimension.

The most salient runtime improvement comes from the ability to find a random farthest witness in $n$9 rounds, ensuring that the overall approach stays within truly subquadratic complexity for the specified graph classes.

Strong Results and Novel Claims

• The paper establishes the first truly subquadratic algorithm for computing the exact diameter and all eccentricities in real-weighted $r$0-minor-free digraphs, a task previously only available for planar graphs or for minor-free graphs in the unweighted setting.
• The algorithm runs in $r$1 time for $r$2-minor-free digraphs.
• The core innovation is the use of random-witness search-to-decision reduction to circumvent the exponential search space due to real edge weights.
• The technique applies broadly to any graph family with both sublinear balanced separators and bounded generalized distance VC-dimension, covering classes such as geometric intersection graphs and polynomial expansion families.

Practical and Theoretical Implications

On the practical side, these results unlock efficient diameter and eccentricity computations for a much broader family of sparse graphs, including those arising from geometric and topological constraints in real-world networks and data analysis.

From a theoretical perspective, the work demonstrates that the range of VC-dimension-based algorithmic techniques extends beyond unweighted or combinatorial settings, provided randomized augmentations to the classic decision procedure using random witnesses. The design of interval merging and sampling subroutines on low-crossing-paths set systems may inspire further practical fine-tuning as well as theoretical extensions. The methodology, especially the synergy between VC-dimension and randomized search, may have applications in designing efficient algorithms for other metric-based problems in structured graphs.

A notable limitation is the algorithm's reliance on the existence of strongly sublinear balanced separators, which excludes certain classes of sparse graphs. Additionally, the current framework does not extend to computation of other distance-based statistics, such as the Wiener index, due to the need for efficiently capturing the full distance distribution rather than just extremal values.

Open directions include removal or weakening of the balanced separator requirement, possibly through more sophisticated decompositions or new low-diameter decomposition techniques adapted to real-weighted graphs. Moreover, innovations in random-witness frameworks may yield efficient diameter algorithms for other distance measures or for more general sparse graph classes.

Conclusion

This work provides a substantial advance in diameter and eccentricity computation for real-weighted graphs, showing that bounded VC-dimension and separator structures, together with random-witness search mechanisms, yield efficient algorithms in settings where none were previously known. The approach broadens the applicability of VC-dimension in algorithmic graph theory, suggesting further exploration in both the design of metric algorithms and the structural analysis of sparse graphs (2607.01926).