distance vc-dimension

This work addresses the challenge of high-dimensional spherical range counting, which is notoriously hindered by the curse of dimensionality, making it difficult to simultaneously achieve accuracy and efficiency. For weighted point sets, we present the first data structure supporting approximate spherical range counting with an arbitrary stretch factor $1+\varepsilon$ ($\varepsilon>0$), maintaining sublinear query time even when the number of points in the ambiguous region, $t_q$, is sublinear. By integrating techniques from approximate range searching, high-dimensional geometric indexing, and query-driven preprocessing, our approach achieves near-linear space complexity $O(n^{1+o(1)})$ and query time $n^{1-\Theta(\varepsilon^4/\log(1/\varepsilon))} + t_q^{\varrho} n^{1-\varrho}$, where $\varrho = \Theta(\varepsilon^2)$. This significantly outperforms existing methods in both theoretical guarantees and practical scalability.

This paper investigates tight bounds on distance preservers, hopsets, and shortcut sets in directed graphs. We introduce the first generic “directed → undirected” reduction framework for distance preservers, enabling systematic lower-bound transfers. Our methodology integrates combinatorial graph theory, structured graph constructions, probabilistic analysis, distance sensitivity arguments, and reduction techniques. Key contributions include: (i) the first Ω(n²⁄⁹) lower bound on the hopbound of O(m)-size hopsets—significantly improving prior results; (ii) a new upper bound of Õ(n⁵⁄⁶p²⁄³ + n) for exact distance preservers on p source-sink pairs; (iii) Ω(n¹⁄²) lower bounds for multiple classes of hopsets and shortcut sets; and (iv) a fundamental separation between polynomial- and arbitrarily-weighted (i.e., arbitrary aspect-ratio) graphs. All bounds are asymptotically tight or represent substantial improvements over the state of the art.

This paper investigates the exact and approximate computation of the Vapnik–Chervonenkis (VC) dimension within the framework of parameterized complexity. For hypergraphs, it establishes that the VC dimension admits both a 1-additive fixed-parameter tractable (FPT) approximation algorithm parameterized by maximum degree Δ and an FPT enumerative algorithm parameterized by dimension d—demonstrating that these are the only structural parameters yielding FPT solvability. Extending to graphs, the authors introduce a generalized VC dimension based on treewidth tw and devise a single-exponential FPT algorithm running in $2^{O(tw)} cdot ext{poly}(n)$ time. Through conditional lower bounds under the Exponential Time Hypothesis (ETH), treewidth-based decomposition, and a hypergraph-to-graph structural mapping, the work establishes fine-grained complexity boundaries for VC dimension computation. It provides the first systematic characterization of the completeness of admissible parameters and delineates the fundamental algorithmic limits for this problem.

Classical computational geometry algorithms often lack sensitivity to structural regularities in geometric inputs, leading to suboptimal performance on partially ordered or structured instances. Method: We introduce *range-partition entropy*, a unified input entropy measure that generalizes structural entropy from sorting to geometric problems—its first such extension. Leveraging this entropy, we design adaptive algorithms for fundamental tasks including 2D extreme points, 2D/3D convex hulls, and visibility queries. These algorithms integrate divide-and-conquer with preprocessing-based sorting to dynamically exploit local order in the input. Results: Our algorithms achieve entropy-sensitive running times—i.e., asymptotic complexity improves as input entropy decreases. Theoretical analysis shows they attain input-dependent optimal or near-optimal bounds for convex hulls and related problems, significantly outperforming traditional worst-case-optimal algorithms on structured inputs.

This work investigates the construction and theoretical limits of sparse navigable graphs over high-dimensional point sets: specifically, whether, under arbitrary distance functions, graphs with sufficiently low average degree exist such that greedy routing always succeeds from any source to any target. Methodologically, the authors integrate high-dimensional geometry, probabilistic analysis—including binomial anti-concentration inequalities—and graph theory, overcoming prior restrictions to low-dimensional or distribution-specific settings. Their contributions are threefold: (i) they establish tight asymptotic bounds on navigability in high dimensions; (ii) they propose a generic construction achieving average degree $O(sqrt{n log n})$; and (iii) they prove a matching $Omega(n^{1/2})$ lower bound—demonstrating that for $O(log n)$-dimensional random point sets, every navigable graph must have average degree at least $Omega(n^{1/2})$. These results provide both foundational theory and practical constructions for high-dimensional nearest-neighbor search.

This study addresses the inefficiency in constructing distance structures for directed graphs by proposing faster algorithms for directed (1+ε)-hopsets, deterministic shortcut sets, and source-wise distance preservers. Methodologically, this work presents the first efficient algorithms that match the latest theoretical bounds, extending them to general graphs via DAG projection techniques while optimizing the computational pipeline through structured hopset analysis and accelerated reduction strategies. By significantly improving the construction speed of directed graph distance structures and achieving state-of-the-art trade-offs between size and hop count, this project provides novel theoretical and practical foundations for shortest path problems in directed graphs.

研究了高维空间中快速度量分解算法，针对ℓ∞和ℓ2空间提出新算法，提高分解效率和参数性能。

This study addresses the lack of efficient algorithms for tournament isomorphism with bounded VC dimension by introducing invariant decomposition and patch tournaments, alongside a balanced recursive algorithm leveraging near-twin structures. The proposed framework achieves isomorphism testing in n^{O(d log d)} time and enables polynomial-time computation of automorphism groups. Furthermore, it establishes that isomorphism for tournaments with bounded chromatic number is solvable in polynomial time. By overcoming existing complexity bottlenecks, this work provides a novel theoretical foundation and efficient algorithmic tools for structured tournament isomorphism problems.

This work investigates parameterized approximation bounds for Partial Set Cover and Maximum Coverage in set systems with bounded VC dimension. By introducing structural parameters such as the shatter function exponent and downward intersection complexity, it establishes the first hardness result showing that Partial Set Cover admits no $(2-\delta)$-approximation in FPT time when VC dimension is at least 7, unless $\text{FPT} = \text{W[1]}$. Conversely, under bounded shatter function exponent, a $(k+1)$-approximation guarantee is recovered and extended to weighted settings, multi-criteria objectives, and matroid constraints. Leveraging parameterized reductions and structural characterizations, the study designs a $2^{O(\Gamma k \log k)} N$-time algorithm for Weighted Partial Set Cover and an EPAS running in $2^{\tilde{O}(kd/\varepsilon)} N^{O(1)}$ time for Weighted CC-MaxSAT under bounded VC dimension.

This work investigates combinatorial bounds for codes in general finite metric spaces, with a focus on conditions under which the classical Gilbert–Varshamov (GV) bound can be surpassed. By modeling codes as independent sets in proximity graphs, the authors develop a generalized GV framework applicable to arbitrary metric spaces and introduce novel concepts such as Ramsey–Sidorenko graphs and independence-forcing graphs. Their analysis demonstrates that local subgraph statistics alone are insufficient to exceed the GV bound; instead, global structural properties of the space are essential. Leveraging tools from graph theory, extremal combinatorics, entropy optimization via KKT conditions, and fractional packing techniques, they derive code bounds for both vertex-transitive and non-edge-transitive graphs, establishing density thresholds for several graph families. In particular, they prove that in Hamming spaces, no improvement over the GV bound is possible using only local information, and provide a tight upper bound based on fractional packing.