Papers

Resting-state functional connectivity (FC) is widely used to classify brain phenotypes and disorders. Most pipelines use the full connectome and seek gains through model design. We instead examine how FC geometry constrains classification and cross-site transfer. Across-subject FC variation concentrates in a small effective subspace, suggesting substantial redundancy in nominal dimensions. Across cohorts, these subspaces may differ in orientation even when their effective ranks are comparable, potentially limiting transfer. Across 2,330 subjects from HCP, ABIDE, and ADHD-200, effective-rank analysis reveals strong spectral concentration. Projection onto leading components at the effective-rank scale recovers most of the full-FC classification performance. In ABIDE, site-specific effective subspaces are weakly aligned, and their principal-angle overlap predicts pairwise transfer after covariate adjustment despite comparable per-site effective ranks. Controlled rotations that alter subspace orientation while preserving the mean and covariance spectrum drive transfer toward chance, whereas displacement-matched label-orthogonal rotations do not. These results identify subspace orientation as a key factor in transfer degradation under controlled perturbations. This study offers a geometric diagnostic of FC generalization and suggests evaluating cross-site harmonization by its ability to align effective subspaces alongside classification accuracy.

We study a collapse process to construct the farthest Voronoi diagram of lines in three dimensions, given a spherical map of the diagram's unbounded features. The collapse process sweeps through the diagram in order of decreasing distance from the farthest lines. It follows the shrinking map, a cell complex on a topological sphere that encodes the locus of points with a fixed farthest distance. We show that, in three dimensions, the collapse process has exactly four non-terminal local event types that change the structure of the shrinking map: deletion, swap, local minimum, and local maximum events; plus one terminal event. This list is complete. We give intrinsic three-dimensional geometric descriptions of the four non-terminal events. First, we classify the two vertex-related events, deletion and swap events, by the spherical convex hull of the four tangent points from a vertex to its four defining lines. Then, we analyze the events related to the local extrema of the distance function along the trisector of three lines. We show that the distance function along a trisector has at most $4$ local maxima and $8$ local minima, and that both bounds are tight. The extrema can be found via a polynomial of degree $12$. As a byproduct, this gives a direct method for finding the smallest sphere tangent to three given lines. At each local extremum, the tangent sphere touches the three lines at points lying on a great circle. The collapse process and the completeness of the event list also apply, under similar general position assumptions, to the farthest Voronoi diagram of convex sites under strictly convex distance functions.

We study exact one-center optimization for axis-parallel segments using axis-parallel squares under endpoint-based coverage, where a segment is \emph{$1$-covered} if the square contains at least one of its endpoints. In the monochromatic problem, we seek a minimum-side-length square that $1$-covers all $n$ input segments. We obtain $O(n\log n)$-time algorithms for both unrestricted and segment-constrained centers. The unrestricted bound matches the known bound implied by the two-representative color-spanning-square problem, whereas the segment-constrained result is new. We also prove matching $\Omega(n\log n)$ lower bounds for both center models in the fixed-order algebraic decision-tree model. In the bichromatic problem, an admissible square must fully contain all $m$ blue segments, minimize the number of red segments with an endpoint in its interior, and, subject to this minimum, maximize its side length within a prescribed bounding box. We give deterministic $O(m+n\log^2 n)$-time and $O(m+mn\log n)$-time algorithms for unrestricted and blue-segment-constrained centers, respectively.

Smart maritime infrastructures provide continuous access to heterogeneous sensing streams, enabling repeated experimentation, digital-twin development, and AI-based maritime services. However, sensing hardware alone is not sufficient for scene-specific model development: historical video streams must also be spatially indexed, contextualized, and reduced to informative subsets for annotation. This paper presents a frame-to-panorama localization and context-aware sampling pipeline for ship detection in historical PTZ maritime video lacking reliable pan, tilt, and zoom metadata. The main contribution is an end-to-end data-curation approach that recovers camera-view information from historical PTZ video and combines it with environmental context and visual diversity to construct compact, scene-specific training sets. Specifically, frames are localized on a reference panorama using SuperPoint and LightGlue, enriched with weather and solar-state metadata, and selected through diversity sampling to preserve variation across camera view and environmental conditions. A second context-aware stage targets under-represented distant-vessel cases near the horizon using tile-level visual embeddings and Gaussian Mixture Model clustering. Applied within the CMMI MDigi-I Smart Marina testbed, the proposed pipeline reduces 40,718 candidate frames to 220 images for annotation, corresponding to a 99.5% reduction. A YOLO26-m detector fine-tuned on this subset achieves a mean AP50 of 94.78% $\pm$ 0.51% and a mean AP50-95 of 75.10% $\pm$ 1.73% under sequence-grouped five-fold cross-validation. These results demonstrate that highly redundant infrastructure video streams can be transformed into compact, spatially and contextually diverse training sets for scene-specific detector adaptation while substantially reducing annotation effort.

We study the \textsc{$k$-Disjoint Paths} problem on a graph embedded on a surface with bounded Euler genus. Given a graph $G$ with $n$ vertices and $k$ vertex pairs embedded on a surface of Euler genus $g$, we present a $2^{O(k^2+g^2)}n$-time algorithm that computes $k$ pairwise vertex-disjoint paths connecting the given vertex pairs if such paths exist. Our approach relies on the decomposition of $G$ into $O(k+g)$ planar subgraphs while bounding the complexity of the boundaries between these subgraphs. This approach enables the use of techniques for compressing linkages in planar graphs. Moreover, our techniques yield two kernels of size polynomial in $k$, $g$, and the treewidth of the graph, and of size $2^{O(k+g)}$. These results extend recent advances on \textsc{$k$-Disjoint Paths} on planar graphs [Cho et al. SODA 2023] and [Włodarczyk and Zehavi FOCS 2023] to surface-embedded graphs.

3D foundation models recover video cameras and geometry in one forward pass, but some of the strongest are up to scale. Joint people-scene reconstruction then requires two missing outputs: metric scale and persistent person identity. We ask whether one up-to-scale foundation representation can support both through lightweight adaptation. Exact metric labels are scarce, but unlabeled in-the-wild video is abundant. We use people in curated web video to initialise the solution: a posed metric body and 2D keypoints give an approximate, closed-form scale pseudo-label. These pseudo-labels pretrain a Scale Readout, which is then fine-tuned together with a lightweight adapter using exact metric supervision from standard real-video training splits. At inference the head predicts metric scale from foundation-model tokens, without the ruler or its teachers. For person identity, we probe the pretrained foundation model alone and find evidence that its intermediate query-key features encode person correspondence across frames. In most evaluated moving-person clips, a mid-layer token prefers that person over the vacated location and other people. A tiny projection reads this correspondence; together with metric pelvis motion and proposal confidence, it drives dustbin-aware Sinkhorn association of per-frame bodies. WildHSR combines both readouts to reconstruct metric cameras, scene and people from monocular video. Each window is predicted feed-forward; analytic association and Sim(3) composition connect windows. On EMDB-2, WildHSR is the first feed-forward method in the published comparison to beat the best optimization-based WA-MPJPE and RTE while leading feed-forward methods on all three world-frame metrics. On RICH, it leads feed-forward people-and-scene methods on WA-MPJPE and W-MPJPE. The complete pipeline runs at 10.1 fps on one GPU.

Approximate convex decomposition (ACD) converts triangle meshes into small sets of convex parts and is a standard preprocessing step for physics simulation, collision detection, and large-scale robot learning. The majority of modern ACD methods produce high-quality decompositions through an expensive search over candidate cutting planes, with per-mesh runtimes of tens of seconds that force game pipelines into overnight bakes and keep articulated-object datasets on CPU clusters for days. Prior work has accelerated isolated stages, most recently VisACD's GPU-based visibility metric, yet the dominant costs – search, mesh cutting, and convex hull construction – have remained on the CPU because their natural decomposition into many small homogeneous phases trails off in a fading last wave at every kernel boundary, and the variable-sized output of each phase forces a host round trip simply to allocate the next launch's input. We address these obstacles by adopting the warp, rather than the thread or thread block, as the unit of algorithm design, an idea introduced in the graph-processing community for a different pathology and which we adapt here to fuse the many heterogeneous phases of a computational-geometry pipeline into single warp-resident kernels, paired with a device-side heap allocator that lets the buffers between fused phases be sized and allocated on the device. Building on this template, we present CuACD (CUDA ACD), the first fully GPU-resident ACD system, together with a suite of reusable GPU components, released as open-source standalone CUDA modules that drop into any search-based ACD pipeline. On the V-HACD benchmark, PartNet-Mobility, and an Objaverse subset, CuACD achieves more than an order of magnitude of speedup over CoACD at matched or better quality.

Existing link-based combinatorial preprocessing methods speed up the computation of persistent homology by removing a vertex or edge only when its link remains a cone. We replace this condition with a quantitative homological certificate. The reduced homology of the filtered link of a generator (a vertex or edge) defines a local obstruction module whose future part describes the effect of deleting that generator. Its barcode certifies either exact deletion or an explicit bound on the bottleneck error, and a conflict colouring extends this guarantee to families of generators. Our implementation, HomTrim, removes an additional 13% to 41% of the input edges beyond domination-only preprocessing and reduces backend persistence time by factors ranging from 1.55 to 4.61 on weighted flag filtrations. The same module also yields regularity diagrams that measure how far a generator can move before becoming visible to homology, together with a multiscale stability result.

This paper presents a representation-aware framework for geometry-based screening of hull-form producibility at early design stages. The proposed signature combines dimensionless total and signed developability deviation with curvature-class area fractions, distributed fields, metric-specific validity, and representation provenance. These descriptors characterize surface features relevant to plate forming and developability, but are not calibrated predictors of fabrication cost, forming effort, or process feasibility. Native IGES/STEP boundary representations (BReps) are evaluated through direct differential geometry and trimmed-domain quadrature, whereas triangulated surfaces use discrete curvature recovery and area-weighted aggregation. Analytical and semi-analytical controls verify the formulation, while matched-face BRep-to-mesh tests assess discrete curvature recovery. Application to DTMB 5415, KCS, JBC, and KVLCC2M shows that curvature intensity and areal extent provide complementary information and that derivative-based outcomes can be representation sensitive. KCS, for example, exhibits approximately 24% greater developability deviation than DTMB 5415, while double-curved regions occupy 72.9% of its valid surface versus nearly the entire DTMB valid surface. The resulting quantities provide an early geometric screening layer for subsequent use as objectives, constraints, surrogate responses, or design-space features. HullProd, the companion open-source software, implements the signature, distributed fields, validity, and provenance.

When designing algorithms for geometric graphs, exploiting structural parameters can lead to significantly improved bounds. Two prominent parameters in this context are $c$-packedness and $\lambda$-low density, both of which locally restrict graph complexity. Parameterized algorithms based on these parameters have been developed for computing well-separated pair decompositions, balanced separators, as well as distance oracles. Nevertheless the practical applicability of algorithms parameterized by $c$ or $\lambda$ remains unclear. While $c$-packed and $\lambda$-low-density graphs have been proposed as realistic models for road networks, the actual parameter values of large real-world instances have so far remained unknown, and existing theoretical guarantees are partially too loose for practical usage. In this paper we first devise scalable implementations for the approximate computation of $c$ and the exact computation of $\lambda$. Our experiments on road networks with millions of edges reveals a significant gap between the two parameters. On the theoretical side we prove that $c\in O(\lambda\sqrt n)$ which complements the known result that $\lambda\in O(c)$. Furthermore we present improved parameterized algorithms for balanced separator computation that reduce the separator size in theory and practice. We also show how to compute a tree decomposition with a width linear in the respective parameterized balanced separator size in polynomial time. This structural result yields a variety of new algorithmic consequences. Among them is an exact distance oracle with query time $O(c)$ for $c$-packed graphs after polynomial-time preprocessing, which improves upon the previous $O(c\log n)$ bound. Our experiments show that the proposed techniques efficiently produce small balanced separators and enable the construction of concise exact distance oracles on large road networks.

A graph is 1-planar if it has a drawing in which every edge is crossed at most once. We show that the smallest cubic non-1-planar graphs have $30$ vertices. Two such graphs are the Tutte-Coxeter graph of girth eight and a graph of girth seven that we call the Byte graph. Every subcubic graph with fewer than $30$ vertices is 1-planar. Our proof is computer-assisted, but directly testing all relevant graphs is impractical. To establish non-1-planarity of the two graphs, we extend a SAT-based solver with a custom clause propagator based on separating cycles and a case split based on graph automorphisms, allowing independent cases to be solved in parallel. To show that all smaller subcubic graphs are 1-planar, we introduce the concept of $k$-flexibility: every set of at most $k$ prescribed edges can remain uncrossed in some 1-planar drawing. We use this property to reconstruct 1-planar drawings of larger graphs from drawings of smaller $k$-flexible graphs. This replaces exhaustive testing of more than forty billion cubic graphs with computations on far fewer graphs of smaller order.

We study the problem of constructing metric spanners in general metric spaces in subquadratic time when given blackbox access to a fast algorithm for batch approximate nearest neighbor search. In particular, we show the following results for any metric space $\mathsf{M} = ([n], \mathsf{d})$ with aspect ratio $\Delta$ admitting a $c$-approximate batch nearest neighbor search algorithm with runtime $\tau_{\mathsf{M}}(n)$, (1) There exists an algorithm that, for any $k \in \mathbb{N}$, constructs an $O(c k)$-distortion spanner with $\tilde{O}(kn^{1+1/2k} \log \Delta)$ edges and runs in time $\tilde{O}(\tau_{\mathsf{M}} \cdot k n^{1/k} \log \Delta)$. (2) Any algorithm that learns at most $o(n^{1+1/k}/k)$ pairwise distances by querying a distance oracle and a blackbox batch nearest neighbor search oracle necessarily incurs $\Omega(c k)$ distortion. Our results entail that (truly) sub-quadratic time algorithms for spanner construction is equivalent to subquadratic time BANN (up to constant-factor losses). As a further application, we use our fast spanner constructions to obtain a fast algorithm for approximating the Wasserstein distance $\mathsf{W}_q$, for all $q > 1$, over any metric space admitting an efficient batch approximate nearest neighbor search algorithm. Together with recent new efficient algorithms for approximate nearest neighbor search in $\ell_p$ spaces, for $p > 2$, our results entail the first subquadratic time algorithms for spanner construction (with the stated size-distortion tradeoff) and $\mathsf{W}_q$ distance approximation over these metric spaces.

We exhibit simple, rotationally symmetric Venn diagrams with 17 curves, with 19 curves and with 23 curves: $n$ Jordan curves carried to one another by rotation through $2\pi/n$, with every one of the $2^n$ regions present and connected and, since the diagrams are simple, every crossing on exactly two curves. Symmetric Venn diagrams exist for every prime number of curves (Griggs, Killian and Savage, 2004), but those diagrams have many curves through a point; simple ones were known only up to 13 curves (Mamakani and Ruskey, 2014). Four 17-curve, twelve 19-curve and six 23-curve diagrams were found by a Metropolis walk on rotation-invariant quadrangulations of the sphere in which regions may temporarily be duplicated, started from the Griggs-Killian-Savage diagram with its multiple crossings resolved. At 23 curves the walk was held for two weeks by duplicated regions near the poles; the two lineages that finished were the first whose $E=92$ states had none. Every diagram is given by a machine-checkable certificate; one certificate each of the 17-, 19- and 23-curve sizes has been verified by a formal proof in Lean 4. All of the diagrams are non-monotone, which is why the crossing-sequence searches that found the 11- and 13-curve diagrams could not have found them.

We introduce a new formulation of the Longest Edge Bisection (LEB) of tetrahedra entirely in sextuple space R6, where tetrahedra are represented by the squares of their edge lengths. This representation renders the LEB refinement equations fully linear and eliminates the need for coordinate-based data structures. A central difficulty in three-dimensional LEB arises when a tetrahedron possesses multiple longest edges, making the refinement rule intrinsically multivalued. We formalize this phenomenon through the notion of Multiform Longest Edge Bisection (MLEB), which systematically explores all admissible longest-edge choices. To encode this multivalued behavior, we introduce the concept of bisection patterns, defined as sequences of sextuple permutations governing the refinement process. We prove that the set of sextuples sharing a common LEB pattern forms a convex region in R6. For structurally significant families of tetrahedra, including the R1+ family and the Liu-Joe family, we show that the infinite refinement tree collapses into a finite directed graph with eight states. Remarkably, both families are governed by the same graph, differing only in their initial state. This directed-graph formulation provides a unified combinatorial description of the refinement process and offers an efficient computational framework for deep iterative LEB analysis.

In this paper, we study the minimum-cost bipartite matching with penalties problem in metric spaces with bounded doubling dimension: Given two disjoint sets $R, B$ in a metric space $\mathcal{M}$ with $|R|+|B|=n$ and a penalty function $p \colon R \cup B \to \mathbb{R}_{\ge 0}$, the goal is to select a set of pairs in $R\times B$ so that every point belongs to at most one pair and the sum of the distances of the selected pairs and the penalties of the points not belonging to any pair is minimized. While near-linear time approximation algorithms are known for the minimum-cost perfect matching problem in geometric settings, no such algorithm was previously known for the penalty setting. We present a randomized algorithm that computes a $(1+\varepsilon)$-approximate minimum-cost bipartite matching with penalties in $O(n \mathrm{poly}(\log n, 1/\varepsilon))$ time with high probability. To the best of our knowledge, this is the first near-linear time approximation algorithm for the problem in the penalty setting.

We study a finite tiling problem, where tiles are unit squares whose four edges are colored with one of two colors. We ask whether a given rectangle admits a perfect rectangular tiling: every cell of the rectangle is occupied by one tile, neighboring edge colors match, and exactly $n_i$ tiles of type $i$ are used, where rotations of the tiles are allowed. Our problem is related to classical Wang tilings, more general finite tile-placement problems, and edge placement puzzles. But in our problem, the multiplicities of the tile types are part of the input and the tile alphabet is fixed and extremely small; thus the complexity of the problem arises from the interaction between the rectangle dimensions and the prescribed tile multiplicities. We provide a comprehensive study of the perfect rectangular tiling problem. For this we consider all classes of subsets of the six possible tile types for two-colored edges, and we characterize for each class whether multiplicities either always allow a perfect rectangular tiling or whether their existence can be decided efficiently.

We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph $H$ with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of $H$. These results extend the recent breakthrough of Bonnet (2026), guaranteeing six pairwise visible points, and come within one visibility edge of the next open case of the big-line-big-clique conjecture.

In the one-round discrete Voronoi game a multiset $V$ of $n$ voters on a line is given; player P places $k$ facilities, player Q then places $\ell$, and each voter is won by the nearer facility, ties going to P. P wins if it keeps at least $n/2$ voters. In the vocabulary of competitive location this is the absolute $(\ell|k)$-centroid problem on a path with unit demands, and the responder's problem is the $(\ell|X_k)$-medianoid, whose closed form on a path – the sum of the $\ell$ largest of at most $2k$ explicit marginals – is due to Spoerhase and Wirth. We record this structure, with complete proofs, and draw two consequences that we believe are new. First, we compute the value of the game against a single responding facility, $\Gamma_{k,1}(V)$, together with an optimal strategy for P, in $O(n\log n)$ time for arbitrary positive real demands and every $k$. This improves the $O(kn\log^2 n)$ bound of Lazar and Tamir for the absolute $(1|k)$-centroid on a path. Second, we study the facility advantage $k^*(\ell)$, the least $k$ for which P wins every instance against $\ell$ facilities. We prove $k^*(\ell)\le 2\ell-1$, exhibit instances proving $k^*(\ell)\ge\ell+1$ for $2\le\ell\le6$ (an exact, computer-assisted proof resting on a half-integer discretisation), determine $k^*(1)=1$ and $k^*(2)=3$, and show that on uniform instances $k=\ell$ already suffices, so the extremal instances are weighted and Q wins them by a single voter. We conjecture $k^*(\ell)=\ell+1$ for all $\ell\ge2$.

In the classical Art Gallery Problem (AGP), guards are placed in a polygon so that together they see every point. The Witness Set Problem (WSP), introduced by Amit, Mitchell, and Packer, is a natural dual to the AGP. In this paper, we study the WSP in weak visibility polygons (WVPs), the simple polygons in which every point is seen from some point of one fixed edge. A witness set is a set of points whose visibility regions are pairwise disjoint, so that no single guard sees two of them. A maximum witness set, therefore, lower-bounds the guard number. Exact polynomial-time algorithms for the WSP are known only for monotone mountains, a proper subclass of WVPs. We give the first exact polynomial-time algorithms for the WSP in WVPs, in two settings. In the Discrete Witness Set Problem (DiscWSP), the witnesses come from a given set of $m$ points, and we find a maximum witness subset in $O(n + m \log(n+m))$ time on an $n$-vertex polygon. The algorithm rests on a structural fact: the visibility intersection graph of a WVP, in which two points are adjacent if their visibility regions intersect, is a trapezoid graph, that is, an intersection graph of trapezoids between two parallel lines. Moreover, the class of these graphs properly contains the interval graphs and the permutation graphs, which may be of independent interest in graph theory. We also prove an $\Omega(n \log n)$ lower bound in the algebraic decision-tree model for instances with $m = \Theta(n)$, so our algorithm for DiscWSP is optimum. In the Continuous Witness Set Problem (ContWSP), a witness may be any point of the polygon, and we give an exact algorithm running in $O(n \log n + \rho^{2}(n + \rho^{2}))$ time, where $\rho$ is the number of reflex vertices.

A ship starts at a point of the plane and moves at unit speed; it has to reach an unknown straight line, of which neither the distance nor the direction is known. The competitive ratio of a path is the supremum, over all lines, of the time at which the line is reached divided by its distance. Baeza-Yates, Culberson and Rawlins conjectured that a logarithmic spiral, with ratio $C_{\mathrm{sp}} = 13.8111351794611\ldots$, is optimal. We give a computer-assisted proof. Paths are arbitrary: the distance from the start and the polar angle may both decrease. The proof lifts the set of found directions to the universal cover of the circle, where unfolding the polar angle can only increase it (Kneser-Poulsen on the line); a bookkeeping inequality with a monotone final source then bounds the covered measure by the reward of a three-state relaxed control problem, in which inward motion is an ordinary control and excursions below the guaranteed disk are impulses. An explicit $C^1$ storage function, a tensor cubic B-spline plus a closed-form term, satisfies the dissipation inequalities of that problem at the spiral's level and is tight only at the spiral; this is verified with about $10^6$ boxes of Arb ball arithmetic, an exact jet and an interval Hessian at the spiral.

Contrastive self-supervised learning has achieved strong performance by learning representations from multiple augmented views of the same image. However, most existing methods construct positive pairs using independently sampled stochastic augmentations, which may alter semantic content and ignore the intrinsic geometry of the data distribution. In this work, we propose OTCLR, an optimal transport-aware framework for contrastive learning representations that generates geometry-consistent positive samples. Instead of directly contrasting two randomly augmented views, we construct intermediate views between the original image and its augmented variants through entropic optimal-transport displacement interpolation. These transport-interpolated samples serve as positive views that better preserve image structure while explicitly modeling spatial distributional geometry. To further promote smooth representation learning, we evaluate auxiliary Sinkhorn regularization terms that encourage transport-interpolated views to remain consistent with their endpoint images. The proposed method can be incorporated into standard contrastive learning pipelines without modifying the encoder architecture. Experiments on multiple benchmark datasets show that our approach improves representation quality and transfer learning performance compared with conventional augmentation-based contrastive learning baselines.

Conflict scanning over synchronized robot paths requires detailed collision checking, potentially across every robot pair at every timestep, and may be repeated many times as conflicts are repaired. We present Multi-Robot SPITE (MR-SPITE), a conservative, motion-segment-based filter for accelerating these scans. MR-SPITE partitions each path into temporal intervals and assigns conservative bounds to each segment. An interval scheduler compares bounds for temporally overlapping motions: disjoint bounds certify the shared window as conflict-free, while unresolved windows are passed to the underlying collision checker. We integrate MR-SPITE into ARC and combine it with VAMP-based collision checking. For 16 Fetch robots, ARC with MR-SPITE achieves a paired median conflict scan speedup of 7.18x and reduces median planning time by 57% relative to the baseline ARC implementation with PRM+VAMP. These results demonstrate that motion-segment bounds complement configuration-level collision acceleration while preserving the behavior of the underlying discretized scanner.

We study the bichromatic line-center problem for $n$ pairs of points in the plane. A feasible solution assigns one point from each pair to the red set $R$ and the other to the blue set $B$. The goal is to minimize $\max\{w^\circ(R),\,w^\circ(B)\}$, where $w^\circ(X)$ denotes the minimum width of a strip enclosing $X$; the midlines of the corresponding optimal strips define the line-centers of $R$ and $B$. We consider several variants induced by orientational constraints on line-centers and provide efficient algorithms for each. For one line-center, which consists of computing a minimum-width strip that contains at least one point from each pair, we give an $O(n^2)$-time algorithm. For two line-centers, we obtain an $O(n)$-time algorithm when both are horizontal, and $\Theta(n\log n)$-time algorithms when the two centers are parallel or when both orientations are prescribed. When exactly one orientation is prescribed, we give an $O(n^2)$-time algorithm. Finally, for the unrestricted case, we present an $O(n^3\log n)$-time algorithm.

Two groups of autonomous, anonymous, and oblivious mobile robots are deployed in the two-dimensional Euclidean plane, each assigned a distinct task. We study a setting where the two groups must simultaneously solve two conflicting pattern formation problems: the gathering problem, where robots gather at a point not known to them a priori, and the circle formation problem, where robots occupy distinct positions on the boundary of a circle. Although each robot knows its own task, it cannot identify other members of its group. A prior solution~\cite{Conflict-1} addressed this problem for asynchronous robots having direction-only axis agreement and global weak multiplicity detection capability available to all robots in both groups. In contrast, in this work, we consider fully disoriented robots without any axis agreement or common chirality. We study the feasibility of a solution to this problem for disoriented robots. We propose a distributed algorithm that solves the problem for semi-synchronous disoriented robots with non-rigid movements. Our proposed algorithm assumes global weak multiplicity detection only for the gathering group, while for the circle formation group, it requires local weak multiplicity detection.

In this paper, we study graph editing problems on geometric intersection graphs. For a tuple $\mathcal{S}=(S_1,\dots,S_n)$ of geometric objects in some Euclidean space, let $G_\mathcal{S}$ be their intersection graph. We study the problem of finding a tuple $D=(d_1,\dots,d_n)$ of movement vectors such that the resulting intersection graph $G_{\mathcal{S}+D}$ (after moving, for every $i \in \{1,\dots,n\}$, object $S_i$ by $d_i$) has a predefined property and the total movement distance $|D|$ is minimum. In the weighted version, we are also given a weight vector $w=(w_1,\dots,w_n)$ with positive entries, and the objective is to minimise the total weighted movement distance $|w \cdot D|$. We first consider the property locally dense, which we define as containment of a $k$-clique. Given $n$ weighted intervals, we solve the problem with respect to this property in $O(k^{1/3} n \log^{1+\varepsilon} n)$ time for any $\varepsilon>0$. We then consider $k$-connectivity for $1\le k \le n-1$. Given $n$ unweighted unit intervals, we solve the problem in $O(n^2 \log n)$ time and, for $k=1$, in $O(n\log n)$ time. For $k=1$, we prove strong NP-hardness on intervals of arbitrary length and on weighted unit disks (with only two distinct weights), and weak NP-hardness on weighted intervals (even when lengths equal weights).

To speed up algorithms on geometric graphs, it is common to approximate the complete Euclidean graph while maintaining certain geometric properties. A (directed) $t$-spanner $G$ for a point set $P$ in the Euclidean space is a (directed) graph such that for every pair of points, the shortest path in $G$ is at most a factor $t$ longer than the Euclidean distance between those points. In this paper, we investigate $t$-spanners that are bounded by certain graph parameters. Let $\kappa$ be a graph parameter. We show that for path-width, branch-width and cut-width there is an $\mathcal{O}(n/k^{d/(d-1)})$-spanner $G$ on $P$ with $\kappa(G)=k$ and that this is asymptotically worst-case optimal. In $\mathbb{R}^2$ we show the same bounds for planar graphs of clique-width or rank-width $k$. In contrast, for tree-depth, we show that there are sets of points for which the dilation cannot be bounded. Therefore, we investigate computing a spanner with tree-depth $k$ and minimum dilation. We show that already for tree-depth $3$ this problem is NP-hard to approximate within any factor strictly less than $\sqrt{2}$, and present an XP-algorithm to compute for a given tree-depth $k$ a graph with dilation at most $2t^*$, where $t^*$ is the minimum dilation. We further extend these results to obtain directed $\mathcal{O}(n/k^{d/(d-1)})$-spanners $G$ with $\kappa(G)=k$ for $\kappa$ being directed tree-width, directed path-width or DAG-width and show that also in the directed case, this is asymptotically worst-case optimal.

Moser's worm problem asks for a planar region of least area containing a congruent copy of every unit arc. We show that the infimum area $\alpha$ among convex universal covers satisfies $0.239\le\alpha\le0.24633\ldots$, reducing the gap between the previous refereed bounds by over $75\%$. For the lower bound, we choose four unit polygonal arcs and prove by finite subdivision that, however they are placed, their convex hull has area at least $0.239$. For the upper bound, we construct a quadrilateral of area $0.24633\ldots$ and prove cover universality by showing that its support inequalities force uncovered arcs to have length greater than one. The full proof is formalized in Lean 4 and verified by the Lean kernel. Code and certificates are available at https://github.com/ethan-keller/moser-worm-improved-bounds.

Let A be a nonempty compact subset of the plane and let A (r) be its parallel set at distance r. We prove that the number of holes of A (r) the number of bounded components of its complement, which is its rst Betti number satises $\infty$ r 0 $\beta$1(A (r) ) dr $\le$ 4050 (diam A) 4 r -3 0 for every r0 > 0, the integrand vanishing for r $\ge$ diam A/ $\sqrt$ 3. The proof rests on Fu's theorem that the critical values of the distance function of a planar compact set form a set of vanishing half-dimensional Hausdor measure, on two lemmas of Rataj, Spodarev and Meschenmoser, and on a square-root summability estimate for the gaps of the critical-value set, of which we give a complete proof. We show by an explicit family of curves two combs facing each other that no analogous bound can hold at a xed radius: a connected curve of bounded length, diameter, oscillation count, parallel-set area and parallel-set perimeter can have arbitrarily many holes at one radius, so the integrated estimate cannot be replaced by a xed-radius bound in terms of these coarse geometric quantities. We also bound the hole count uniformly in the radius by the number of components of local maxima of the distance function, and record a bound on the boundary length of a parallel set by its area. The results supply the deterministic input for limit theorems on the persistent homology of the Wiener sausage.

Rigid-body interpenetration frequently occurs in procedurally assembled and generated scenes and must be removed before downstream applications such as physical simulation. We present S4R (Scaling for Rigid-Body Interpenetration Resolution), a scale-continuation method for static interpenetration repair. S4R first uniformly shrinks each body about a fixed reference center to a small initial scale, at which the layout is penetration-free, and then restores full scale through a sequence of minimum-norm convex contact quadratic programs (QPs) that target the linearized separation margin during continuation. Resolution thereby replaces one deep correction with a sequence of shallow-contact subproblems. A conservative scale-event bound and frozen-witness gap predictions cut the number of exact mesh queries; the continuation then ends with a full-scale evaluator check and bounded tail refinement. We evaluate S4R on Kubric, HY3D-Bench, and Thingi10K using a shared mesh-level evaluator and a unified per-scene timing protocol. In the main comparisons on all three benchmarks, up to N=5000 bodies, S4R reaches zero reported penetration with displacement that stays small and nearly independent of scene size, and at the lowest wall time within each hardware tier among the compared methods. A GPU implementation extends these results to large-scale scenes. Our code and data can be found on our project page: https://frank-zy-dou.github.io/projects/S4R/index.html.

We study the space complexity of diameter estimation for a set of points in Euclidean space in the dynamic (turnstile) streaming model. The seminal work of Indyk (SODA 2003) gives a $c$-approximation to the Euclidean diameter of $n$ vectors using $n^{O(1/c^2)}$ space. Our main contribution is giving an essentially matching lower bound. Any dynamic streaming algorithm which can $c$-approximate the diameter of $n$ Euclidean vectors must use $n^{\tilde{\Omega}(1/c^2)}$ space.