derive minimum distance bounds

This paper investigates the diameter problem for families of permutations defined by constraint graphs (directed acyclic graphs) under the ℓ∞ and Kendall–Tau metrics. The central result establishes a necessary and sufficient condition: the Kendall–Tau diameter achieves its combinatorial upper bound if and only if the partial order induced by the constraint graph has dimension at most two—thereby forging the first exact link between metric extremality and poset dimension. Building on this characterization, we devise an explicit algorithm to construct maximally distant permutation pairs. As corollaries, we obtain closed-form expressions for the metric diameters of classical structures including descent sets and Hessenberg varieties. Our approach integrates tools from poset theory, permutation combinatorics, and graph theory, enabling efficient computation. The results provide new theoretical foundations and algorithmic tools for combinatorial coding, fault-tolerant ranking, and metric embedding.

This paper addresses the Lipschitz decomposition problem in finite ℓₚ metric spaces. For an n-point set, it establishes— for the first time—the optimal splitting parameter bound β = O(log^{1−1/p} n) for p > 2, fully resolving a long-standing open question posed by Naor (SODA 2017); it also significantly improves the best-known bounds for 1 < p ≤ 2. Methodologically, the work integrates probabilistic arguments, geometric functional analysis, and metric embedding theory, introducing novel extensions of hierarchical random partitions and sparse covers. The results advance high-dimensional geometric spanner construction efficiency and yield the first deterministic, logarithmically near-optimal guarantee on compression rate for distance labeling schemes—bridging deep theoretical insights with practical algorithmic impact.

This study addresses the parameterized Metrical Service Systems (MSS) problem, where request types are restricted to a known set of $m$ kinds. By modeling the problem via interval covering and employing a primal-dual approach, the authors design deterministic algorithms and establish matching adversarial lower bounds. On weighted star metrics, they achieve the first $O(m)$-competitive deterministic algorithm, matching the known randomized lower bound. On hierarchically separated trees (HSTs), they prove that no constant-competitive algorithm exists when $m \geq 4$, while presenting an $O(1)$-competitive deterministic algorithm for the case $m = 2$. This work resolves several open questions posed by Bubeck and Rabani, fully characterizing the performance limits of parameterized MSS on these two fundamental metric spaces.

This study addresses the problem of computing a minimum-sized well-separated pair decomposition (minWSPD) in Euclidean spaces of dimension two and higher, which is known to be intractable to solve exactly and lacks efficient approximation algorithms. To overcome this challenge, the authors introduce a novel variant of pair decomposition that relaxes the conventional diameter constraints on subsets. This approach achieves a size bound of $O(n/\varepsilon \cdot \log n)$ in general metric spaces—significantly improving upon the quadratic bound of classical WSPDs—and further refines it to $O(d \cdot n/\varepsilon \cdot \log(1/\varepsilon))$ in $\mathbb{R}^d$. By leveraging doubling dimension, Euclidean geometric structure, and careful approximation design, this work presents the first constant-factor approximation algorithms for minWSPD in low-dimensional Euclidean and doubling metric spaces.

This paper investigates the algorithmic information preservation of Euclidean distances and orthogonal projections in the plane under finite-precision computation. Using finite-precision Kolmogorov complexity and algorithmic information theory, we introduce a proxy-point selection strategy and approximate conditioning techniques. We prove that for any pair of points $x, y$, both the distance $|x - y|$ and the projection coordinate $p_e x$ onto any unit direction $e$ retain at least half the algorithmic information content of the origin. As a consequence, we establish a new lower bound on the Hausdorff dimension of pinned distance sets: if $dim_H E leq 1$, then $sup_{x in E} dim_H(Delta_x E) geq frac{3}{4}dim_H E$. Furthermore, we extend Bourgain’s theorem on exceptional directions for orthogonal projections to all sets admitting optimal Hausdorff oracles, thereby bridging geometric measure theory and algorithmic information theory within a unified computational framework.

This work addresses the optimization of finite parameters in constructions related to counterexamples of the Erdős unit distance conjecture, aiming to improve the exponent in the lower bound on the number of unit distances determined by planar point sets. We formulate the parameter selection in Sawin’s explicit lower-bound proof as a nonlinear integer programming problem and introduce a lightweight, reproducible integer optimization framework. This framework integrates deterministic greedy strategies, a tailored integer evolutionary algorithm, and discrete recombination operators, enhanced by rational-number encoding and a repair mechanism. Implemented on standard hardware, our approach yields a new certificate δ = 0.0152628688…, establishing that for all sufficiently large n, u(n) > n^{1.0152}, thereby improving upon Sawin’s original result.

This study addresses the problem of maximizing the density of unit distance graphs in the rational plane. To overcome the limitations of existing constructions—often constrained by conventional grid structures and unable to surpass known theoretical lower bounds—the authors propose a local breadth-first search algorithm tailored to the rational plane. This approach dispenses with fixed grid constraints and efficiently generates universal unit distance graphs within bounded finite regions. The resulting graphs exhibit significantly higher density than recent theoretical lower bounds, achieving a scaling exponent that improves upon the current best-known results. The method thus offers both a novel perspective and an effective computational tool for advancing lower-bound constructions in this domain.

This work investigates how to accelerate parallel connectivity and shortest-path algorithms by efficiently augmenting a directed acyclic graph with transitive-closure edges to substantially reduce its diameter. The authors introduce a structural criterion called “certified shortcut edges,” which precisely characterizes the set of shortcut edges constructible by any near-linear-time algorithm. Leveraging this criterion in conjunction with combinatorial graph theory and complexity lower-bound techniques, they improve the known diameter lower bound from $n^{2/9 - o(1)}$ to $n^{1/4 - o(1)}$, establishing that no near-linear-time algorithm can achieve a smaller diameter. This result significantly strengthens existing theoretical limits and provides a robust foundation for the design of future efficient shortcut-edge construction methods.

This study investigates the parameterized complexity of the Min-Sum-Radii clustering problem under graph-induced metrics. Although the problem is known to be NP-hard, its tractability with respect to various structural graph parameters remained unclear. We establish that the problem remains W[1]-hard even when parameterized jointly by the vertex cover number and the number of clusters, and we further demonstrate its computational intractability on cliques and complete bipartite graphs. On the positive side, we design a fixed-parameter tractable (FPT) algorithm showing that the problem becomes solvable in FPT time when parameterized by the sum of treewidth and the target cost. This work systematically delineates the complexity landscape of Min-Sum-Radii across diverse graph metrics and parameter combinations.

This work investigates the construction of structurally decomposable representations for planar graphs with annotated vertex sets, aiming to support efficient parameterized algorithms—such as those for Steiner Tree—on H-minor-free graphs. By integrating recent advances in graph minor theory with novel techniques tailored to annotated vertex sets, the paper establishes, for the first time, a structural theorem for vertex sets of bounded twin-width, demonstrating that all associated parameters admit polynomial bounds. This result is further extended to apex-minor-free graphs, thereby providing a robust structural foundation and practical toolkit for designing parameterized algorithms in these graph classes.