arXiv subjects

Jack Spalding-Jamieson

Publications and source records attributed to Jack Spalding-Jamieson.

18 recordsLinked to original sources

arXiv2026

Flow-Optimal Separators for Minor-Free and Induced-Minor-Free Graphs in Linear Time via Follow the Regularized Leader

We give a new algorithm for finding balanced separators in minor-free and induced-minor-free graphs. A key technical idea is a new connection to online learning: we formulate a continuous, non-convex optimization problem and solve it using the Follow the Regularized Leader (FTRL) framework. Our implementation of each FTRL step yields a purely combinatorial algorithm that assigns and adjusts vertex weights over multiple rounds. For graphs with $n$ vertices and $m$ edges, we obtain the following results. (1) For $H$-induced-minor-free graphs, we find a balanced separator of size $O(\log(|V(H)|)\sqrt{|V(H)|+|E(H)|}\sqrt{m})$ in expected linear time for fixed $H$. As a corollary, we find balanced separators of size $O(\sqrt{m})$ in string graphs in expected linear time, without requiring string representations. (2) For $K_h$-minor-free graphs, we find a balanced separator of size $O(h\log h\sqrt{n})$ in expected linear time for fixed $h$. Our algorithm unifies two long lines of work on separators in minor-free graphs, dating back to Alon, Seymour, and Thomas (1990): one optimizing separator size and the other optimizing running time. It also gives a natural interpretation of the vertex weighting scheme in a recent linear-time algorithm by Bonnet, Korhonen, Le, Li, and Masarik (2026). The separator bounds for both graph classes are flow-optimal: techniques based on flow-cut duality, including ours, cannot yield balanced separators of asymptotically smaller size.

cs.DS↗

arXiv2026

$K_h$-Minor-Free Graphs Have Balanced Separators of Size $O(h\sqrt{n})$

We show that every $K_h$-minor-free graph with $n$ vertices has a balanced separator of size $O(h\sqrt{n})$ for any parameter $h \in \mathbb{N}$, resolving the conjecture posed by Alon, Seymour, and Thomas [AST90] in 1990. The proof is elementary, building on the combinatorial framework by Le [Le26]. A key new idea of the proof is the existence of a short hitting walk on vertex expanders: given $k$ large (not necessarily disjoint) subsets of vertices, there is a walk of length $O(k/ψ(G))$ hitting every subset, where $ψ(G)$ is the vertex expansion. This technical result substantially improves on an analogous lemma by Alon, Seymour, and Thomas [AST90] in an important parameter setting, where the previous lower bound [Kot00] does not apply.

math.CO↗

arXiv2026

Optimal Average Distortion for Non-Expansive Embeddings of Euclidean Metrics into the Line: With applications to cut-matching games and vertex expansion

We show that any $n$-point Euclidean metric admits a non-expansive embedding into the line with average distortion at most $O\left(\sqrt{\frac{\log n}{\log\log n}}\right)$, matching a known lower bound. We also present two applications of this result. First, we give a cut-player strategy with an optimal spectral guarantee for the undirected cut-matching game. Second, we give an improved spectral approximation for vertex expansion in high-degree graphs.

cs.DS↗

arXiv2026

Fast Approximation Algorithms for Balanced Edge and Vertex Separators

We present a family of fast pseudo-approximation algorithms for the minimum balanced element cut problem in a graph, which includes vertex and edge separators. Given a graph $G=(V,E)$ with polynomially-bounded finite integer element weights with $n$ vertices and $m$ edges, and a (constant) balance parameter $c\in(0,1/2)$, where $G$ has some (unknown) $c$-balanced element cut of size $OPT_c$, we give a (Monte-Carlo randomized) algorithm running in $O(n^{\varepsilon}m^{1+o(1)})$ time that produces a $Θ(1)$-balanced element cut of size $O(OPT_c\cdot\sqrt{(\log n)/\varepsilon})$ for any value $\varepsilon\in[Θ(1/\log(n)),Θ(1)]$. Our algorithms are based on a semidefinite programming (SDP) relaxation of the problem. A standard approach to solving an SDP relaxation for this kind of problem would use the primal-dual Matrix Multiplicative Weights Update framework of Arora and Kale, but this framework does not seem to apply, since we cannot express all of our SDPs primal variables explicitly. Instead, we present a new MMWU framework for our problem that allows us to represent some of our variables implicitly. We also extend our framework to a general "Partial-variable MMWU framework", which we expect will be useful for future work. Our oracle for the balanced element cut problem within our framework uses $O(n^{\varepsilon}\text{polylog}(n))$ almost-linear time maximum-flow computations, and would be sped up if the time complexity of maximum-flow improves. For graphs with padding bound $β$, another oracle gives an $O(\sqrtβ/\varepsilon)$ approximation in $m^{1+o(1)}n^{\varepsilon}\operatorname{polylog}(n)$ time. It is known that $β\le O(\log h)$ for $K_h$-minor-free graphs (and in particular $β\le O(\log n)$).

cs.DS↗

arXiv2026

Revisiting separators in string graphs, region intersection graphs, and induced-minor-free graphs

String graphs are intersection graphs of continuous curves in the plane. In 2010, Fox and Pach proved that string graphs with $m$ edges admit balanced vertex separators of size $\mathcal{O}(m^{3/4}\sqrt{\log m})$ and conjectured that this bound can be improved to $\mathcal{O}(\sqrt{m})$, which would be optimal. After partial progress, Lee claimed in 2017 to prove this conjecture, and more generally to prove analogous separator theorems for region intersection graphs over minor-closed graph families. A key intermediate step in Lee's approach was a padded decomposition scheme for region intersection graphs. In 2024, Korhonen and Lokshtanov extended Lee's approach to the broader setting of graph classes excluding a fixed graph as an induced minor. These results have since found applications in a wide range of works. Later in 2024, a flaw was discovered in the construction of Lee's padded decomposition scheme, invalidating Lee's as well as Korhonen and Lokshtanov's separator results, and many works relying on them. In this work, we prove that string graphs with $m$ edges admit balanced separators of size $\mathcal{O}(\sqrt{m})$, and more generally obtain analogous $\mathcal{O}(\sqrt{m})$ bounds for region intersection graphs over fixed minor-closed graph families and for graphs excluding a fixed graph as an induced minor, thereby repairing the vast majority of the affected works. Furthermore, our balanced separators can be found in polynomial time. Our proof uses a different approach from Lee's and does not rely on padded decompositions. Since padded decompositions have also found applications independent of the separator theorems, particularly for string graphs, we also prove a padded decomposition result for region intersection graphs that is optimal for string graphs. We have formalized both the balanced separator and the padded decomposition results in Lean 4 with mathlib.

math.CO↗

arXiv2026

Reweighted Spectral Partitioning Works: A Simple Algorithm for Vertex Separators in Special Graph Classes

We establish that a practical polynomial-time algorithm obtains near-optimal $\frac23$-balanced vertex separators for a number of graph classes, including genus-$g$ graphs, $K_h$-minor-free graphs, and geometric graphs. For many of these classes, our polynomial-time separator bounds improve on previous work. Moreover, unlike previous work, all these separators are produced by a single unified algorithm that has no specialization for the graph class, and this algorithm admits a scalable implementation with good empirical results. Furthermore, our approach is robust to graphs falling just outside these classes, making it appropriate for practical applications like road networks. A key distinction of our approach, compared to previous work, is the disentanglement of our algorithm and its analysis for specific graph classes. Our algorithm is based on a newly improved form of a recent Cheeger-style inequality relating the value of a special semidefinite program to vertex expansion. Our analysis then only has to directly bound the value of this semidefinite program for each graph class, and constructions for these bounds need not be algorithmic. Finally, we obtain several results of independent interest. These include an improved (existential) separator theorem for intersection graphs of $k$-ply $d$-dimensional balls, an improved bound on the Fiedler value of genus-$g$ graphs, and a complete spectral proof of the planar separator theorem.

cs.DS↗

arXiv2026

On Eigenvalue Bounds for Bounded Genus Graphs and Minor-Free Graphs

In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $Δ$, we show that the second-smallest eigenvalue of its Laplacian matrix satisfies $λ_2(L_G)\lesssimΔ\frac g n$. We also obtain an improved eigenvalue bound for $K_h$-minor-free graphs of $λ_2(L_G)\lesssimΔ\frac{h^2(\log h)^2}n$. In fact, our results directly prove much stronger results for reweighted eigenvalues, including higher reweighted eigenvalues. As a consequence, we obtain bounds not just on Laplacian eigenvalues, but also on normalized Laplacian eigenvalues and Steklov eigenvalues. Our results for genus-$g$ graphs are optimal for all of these kinds of eigenvalues, while our results for $K_h$-minor-free graphs are optimal up to $\log(h)$ factors. Our techniques for genus-$g$ graphs bootstrap bounded-degree bounds of normalized eigenvalues for entire classes to bounds for reweighted eigenvalues for the same classes without the bounded-degree limitation, while our techniques for $K_h$-minor-free graphs generalize an argument of Korhonen and Lokshtanov, making use of the Lovász local lemma.

math.CO↗

arXiv2026

Subquadratic Approximation Algorithms for Separating Two Points with Objects in the Plane

The (unweighted) point-separation problem asks, given a pair of points $s$ and $t$ in the plane, and a set of candidate geometric objects, for the minimum-size subset of objects whose union blocks all paths from $s$ to $t$. Recent work has shown that the point-separation problem can be characterized as a type of shortest-path problem in a geometric intersection graph within a special lifted space. However, all known solutions to this problem essentially reduce to some form of APSP, and hence take at least quadratic time, even for special object types. We improve the conditional quadratic lower bounds for this problem, but our main results are positive: We bypass this barrier by providing subquadratic algorithms to produce solutions of size $\text{OPT}+1$ or $(1+\varepsilon)\text{OPT}+1$. Our algorithms are fundamentally different from the APSP-based approach. In particular, we give Monte Carlo randomized additive $+1$ approximation algorithms running in $\widetilde{\mathcal{O}}(n^{\frac32})$ time for disks, axis-aligned line segments and constant-complexity rectilinear polylines, and $\widetilde{\mathcal{O}}(n^{\frac{11}6})$ time for line segments and constant-complexity polylines. We will also give deterministic multiplicative-additive approximation algorithms that, for any value $\varepsilon>0$, guarantee a solution of size $(1+\varepsilon)\text{OPT}+1$ while running in $\widetilde{\mathcal{O}}\left(n/\varepsilon\right)$ time for disks, axis-aligned line segments and constant-complexity rectilinear polylines, and $\widetilde{\mathcal{O}}\left(n^{4/3}/\varepsilon\right)$ time for line segments and constant-complexity polylines.

cs.CG↗

arXiv2026

The Presort Hierarchy for Geometric Problems

Many fundamental problems in computational geometry admit no algorithm running in $o(n \log n)$ time for $n$ planar input points, via classical reductions from sorting. Prominent examples include the computation of convex hulls, quadtrees, onion layer decompositions, Euclidean minimum spanning trees, KD-trees, Voronoi diagrams, and decremental closest-pair. A classical result shows that, given $n$ points sorted along a single direction, the convex hull can be constructed in linear time. Subsequent works established that for all of the other above problems, this information does not suffice. In 1989, Aggarwal, Guibas, Saxe, and Shor asked: Under which conditions can a Voronoi diagram be computed in $o(n \log n)$ time? Since then, the question of whether sorting along TWO directions enables a $o(n \log n)$-time algorithm for such problems has remained open and has been repeatedly mentioned in the literature. In this paper, we introduce the Presort Hierarchy: A problem is 1-Presortable if, given a sorting along one axis, it permits a (possibly randomised) $o(n \log n)$-time algorithm. It is 2-Presortable if sortings along both axes suffice. It is Presort-Hard otherwise. Our main result is that quadtrees, and by extension Delaunay triangulations, Voronoi diagrams, and Euclidean minimum spanning trees, are 2-Presortable: we present an algorithm with expected running time $O(n \sqrt{\log n})$. This addresses the longstanding open problem posed by Aggarwal, Guibas, Saxe, and Shor (albeit randomised). We complement this result by showing that some of the other above geometric problems are also 2-Presortable or Presort-Hard.

cs.CG↗

arXiv2025

Separating Two Points with Obstacles in the Plane: Improved Upper and Lower Bounds

Given two points in the plane, and a set of "obstacles" given as curves through the plane with assigned weights, we consider the point-separation problem, which asks for the minimum-weight subset of the obstacles separating the two points. A few computational models for this problem have been previously studied. We give a unified approach to this problem in all models via a reduction to a particular shortest-path problem, and obtain improved running times in essentially all cases. In addition, we also give fine-grained lower bounds for many cases.

cs.CG↗

arXiv2025

The Analytic Arc Cover Problem and its Applications to Contiguous Art Gallery, Polygon Separation, and Shape Carving

We show the following problems are in $\textsf{P}$: 1. The contiguous art gallery problem -- a variation of the art gallery problem where each guard can protect a contiguous interval along the boundary of a simple polygon. This was posed at the open problem session at CCCG '24 by Thomas C. Shermer. 2. The polygon separation problem for line segments -- For two sets of line segments $S_1$ and $S_2$, find a minimum-vertex convex polygon $P$ that completely contains $S_1$ and does not contain or cross any segment of $S_2$. 3. Minimizing the number of half-plane cuts to carve a 3D polytope. To accomplish this, we study the analytic arc cover problem -- an interval set cover problem over the unit circle with infinitely many implicitly-defined arcs, given by a function.

cs.CG↗

arXiv2025

Slant/Gokigen Naname is NP-complete, and Some Variations are in P

In this paper we show that a generalized version of the Nikoli puzzle Slant is NP-complete. We also give polynomial time algorithms for versions of the puzzle where some constraints are omitted. These problems correspond to simultaneously satisfying connectivity and vertex degree constraints in a grid graph and its dual.

cs.DM↗

arXiv2025

Scalable k-Means Clustering for Large k via Seeded Approximate Nearest-Neighbor Search

For very large values of $k$, we consider methods for fast $k$-means clustering of massive datasets with $10^7\sim10^9$ points in high-dimensions ($d\geq100$). All current practical methods for this problem have runtimes at least $Ω(k^2)$. We find that initialization routines are not a bottleneck for this case. Instead, it is critical to improve the speed of Lloyd's local-search algorithm, particularly the step that reassigns points to their closest center. Attempting to improve this step naturally leads us to leverage approximate nearest-neighbor search methods, although this alone is not enough to be practical. Instead, we propose a family of problems we call "Seeded Approximate Nearest-Neighbor Search", for which we propose "Seeded Search-Graph" methods as a solution.

cs.LG↗

arXiv2024

Morphing Planar Graph Drawings via Orthogonal Box Drawings

We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an $n$-vertex planar graph and two planar straight-line drawings of the graph on an $O(n) \times O(n)$ grid. The planarity-preserving morph is composed of $O(n)$ linear morphs between successive pairs of drawings, each on an $O(n) \times O(n)$ grid with a constant number of bends per edge. The algorithm to compute the morph runs in $O(n^2)$ time on a word RAM model with standard arithmetic operations -- in particular no square roots or cube roots are required. The first step of the algorithm is to morph each input drawing to a planar orthogonal box drawing where vertices are represented by boxes and each edge is drawn as a horizontal or vertical segment. The second step is to morph between planar orthogonal box drawings. This is done by extending known techniques for morphing planar orthogonal drawings with point vertices.

cs.CG↗

arXiv2024

Carving Polytopes with Saws in 3D

We investigate the problem of carving an $n$-face triangulated three-dimensional polytope using a tool to make cuts modelled by either a half-plane or sweeps from an infinite ray. In the case of half-planes cuts, we present a deterministic algorithm running in $O(n^2)$ time and a randomized algorithm running in $O(n^{3/2+\varepsilon})$ expected time for any $\varepsilon>0$. In the case of cuts defined by sweeps of infinite rays, we present an algorithm running in $O(n^5)$ time.

cs.CG↗

arXiv2023

Conflict Optimization for Binary CSP Applied to Minimum Partition into Plane Subgraphs and Graph Coloring

CG:SHOP is an annual geometric optimization challenge and the 2022 edition proposed the problem of coloring a certain geometric graph defined by line segments. Surprisingly, the top three teams used the same technique, called conflict optimization. This technique has been introduced in the 2021 edition of the challenge, to solve a coordinated motion planning problem. In this paper, we present the technique in the more general framework of binary constraint satisfaction problems (binary CSP). Then, the top three teams describe their different implementations of the same underlying strategy. We evaluate the performance of those implementations to vertex color not only geometric graphs, but also other types of graphs.

cs.CG↗

arXiv2021

Coordinated Motion Planning Through Randomized k-Opt

This paper examines the approach taken by team gitastrophe in the CG:SHOP 2021 challenge. The challenge was to find a sequence of simultaneous moves of square robots between two given configurations that minimized either total distance travelled or makespan (total time). Our winning approach has two main components: an initialization phase that finds a good initial solution, and a $k$-opt local search phase which optimizes this solution. This led to a first place finish in the distance category and a third place finish in the makespan category.

cs.CG↗

arXiv2020

Angle Covers: Algorithms and Complexity

Consider a graph with a rotation system, namely, for every vertex, a circular ordering of the incident edges. Given such a graph, an angle cover maps every vertex to a pair of consecutive edges in the ordering -- an angle -- such that each edge participates in at least one such pair. We show that any graph of maximum degree 4 admits an angle cover, give a poly-time algorithm for deciding if a graph with no degree-3 vertices has an angle-cover, and prove that, given a graph of maximum degree 5, it is NP-hard to decide whether it admits an angle cover. We also consider extensions of the angle cover problem where every vertex selects a fixed number $a>1$ of angles or where an angle consists of more than two consecutive edges. We show an application of angle covers to the problem of deciding if the 2-blowup of a planar graph has isomorphic thickness 2.

cs.CG↗