manage overlapping partitions

Existing simplex partitioning algorithms rely on restrictive geometric assumptions, limiting their applicability to diverse set systems and hindering generality in data structures such as range searching. This paper introduces the first generalization of simplex partitioning to arbitrary abstract set systems, proposing a geometry-agnostic greedy heuristic that minimizes the crossing number to construct low-complexity partitions. The method provides theoretical guarantees on partition quality while demonstrating empirical robustness across both geometric and non-geometric set systems—including hypergraphs, relational databases, and combinatorial families. An open-source implementation ensures reproducibility and facilitates practical deployment. Our core contributions are threefold: (i) eliminating geometric prerequisites for simplex partitioning; (ii) establishing a unified, model-agnostic partitioning framework; and (iii) providing an efficient, analyzable, and implementable construction paradigm with provable approximation bounds.

This paper addresses the problem of partitioning a convex polygon into the minimum number of subpolygons, subject to a prescribed width constraint and a fixed set of admissible directions. We establish an optimal partitioning theory tailored to width constraints, proving its intrinsic connection to Bang’s conjecture and revealing a key structural property: for any convex polygon, there exists a direction in the given set such that a unidirectional parallel cut sequence achieves the globally optimal partition. Leveraging orthogonal projection analysis, directional monotonicity arguments, and convexity-preserving structural characterization, we devise the first linear-time optimal algorithm. Crucially, our method avoids exhaustive direction enumeration—distinguishing it from prior approaches—and achieves asymptotically superior efficiency. The algorithm has direct practical implications in computational geometry, VLSI floorplanning, and motion planning, where width-constrained decomposition is essential.

This paper resolves the long-standing “minimum star-shaped partition of a simple polygon” problem—open since 1981—by covering a given simple polygon with the fewest non-overlapping star-shaped subpolygons, allowing Steiner points. The proposed method integrates geometric decomposition, visibility graph optimization, dynamic programming, and structural analysis of star kernels, constructing the DP state space over triangulations. It yields the first exact polynomial-time algorithm applicable to arbitrary simple polygons, overcoming prior restrictions to monotone or orthogonal polygons and eliminating the requirement to forbid Steiner points. The algorithm runs in O(n⁹) time, a substantial improvement over exponential brute-force approaches. This theoretical breakthrough enables direct applications in CNC pocket milling, motion planning, and shape parameterization, where minimal star-shaped decompositions are essential for efficient toolpath generation, collision-free navigation, and domain mapping.

This study addresses the problem of partitioning a polygon into the minimum number of strips of width at most 1, aligned with a given orthogonal direction, and producing a compact representation of the optimal partition. It introduces, for the first time in this domain, the Clarke–Cormack–Burkowski lattice-theoretic framework, modeling the problem via interval antichains and combining meet/join operations with dynamic programming to devise an input-sensitive optimal algorithm. For convex polygons, the approach achieves an O(log n)-time decision version and an O(h log(1 + n/h))-time reporting version, where h is the number of strips in the optimal solution. For both simple and self-overlapping polygons, it attains O(n log n) time complexity, while establishing matching lower bounds of Ω(n) and Ω(n log n), respectively, thereby yielding tight complexity characterizations for all three polygon classes.

This paper addresses the segmentation optimization problem on edge-colored graphs: partitioning a graph into color-connected components such that each color appears at most once per component. We formulate three variants—minimizing the number of components, maximizing monochromatic isolation, and others—and propose, for the first time, an exact branch-and-cut algorithm based on integer nonlinear programming with standard linearization. Our method introduces novel contributions: (i) custom valid inequalities; (ii) tight variable bounds; (iii) effective preprocessing rules; (iv) a warm-start strategy; and (v) bound-based pruning to accelerate convergence. Evaluated on benchmark instances, the solver efficiently handles medium-scale problems and constitutes the first scalable exact approach for this problem class. It provides a new computational tool for applications in community detection, cybersecurity, and bioinformatics—particularly in analyzing colored structural patterns.

This study addresses the geometric maximum coverage problem: given a set of geometric objects and a set of points (or a region), select at most $k$ objects to maximize the number of covered points (or total covered volume). By integrating shallow-cell complexity analysis, VC-dimension theory, parameterized algorithms, and geometric decomposition techniques, the work achieves the first improvements over the classical $1 - 1/e$ approximation ratio for various geometric objects—including pseudo-disks, fat rectangles, and same-size fat triangles—under the assumption of linear 2-shallow cell complexity or constant VC-dimension. It also develops more efficient parameterized approximation schemes for small $k$ and establishes an EPTAS for the continuous variant. Furthermore, the paper proves APX-hardness and the infeasibility of PTAS under certain settings.

This study addresses the p-regions problem in geographic zoning, which involves partitioning a region into p connected and attribute-homogeneous subregions—a task known to be NP-hard. To tackle this challenge, we propose a novel integer linear programming model, ER-S, enhanced with specialized subtour elimination inequalities that strengthen connectivity constraints. Building upon structural insights from the k-partitioning problem, we further develop the ER-S-Tree model, which exhibits superior polyhedral strength. Computational experiments demonstrate that our approach significantly outperforms existing methods in both computational efficiency and provable optimality, successfully solving large-scale instances for several European countries for the first time.

This work addresses the efficient computation of intersection areas between multiple circles and complex polygons, a common challenge in applications such as wireless sensor networks. The authors propose a boundary-focused adaptive quadtree algorithm that integrates curvature- and multiplicity-guided dynamic sampling, Green’s theorem–based analytical integration, and an enhanced Monte Carlo subsampling strategy. By concentrating computational resources in geometrically intricate regions and enforcing a minimum sample constraint to bound estimation error, the method achieves high accuracy with controlled complexity. Theoretical analysis shows that the algorithm guarantees an $O(\varepsilon)$ error bound while reducing computational complexity to $O(1/\varepsilon^{3/2})$, outperforming conventional Monte Carlo and uniform grid approaches. Experimental results on both synthetic and real-world polygons demonstrate significantly lower relative errors and strong parameter robustness, confirming its suitability for practical coverage estimation tasks.

Existing contention resolution schemes struggle to control lower-tail probabilities, limiting their applicability to optimization problems with coverage constraints. This work addresses this gap by introducing a novel property—strong λ-boundedness—for the Adamczyk–Włodarczyk random-order contention resolution scheme and formulating a sequential selection process model. This framework yields, for the first time, dimension-free lower-tail bounds independent of the ground set size. Leveraging this analysis together with concentration inequalities and matroid-constrained optimization techniques, the authors improve the approximation ratio for the k-matroid intersection coloring problem to O(k log k) and design the first bicriteria approximation algorithm for monotone submodular maximization that simultaneously handles both covering and packing constraints.