[
  {
    "slug": "3-colorability-of-arrangements-of-great-circles",
    "title": "3-Colorability of Arrangements of Great Circles",
    "aliases": [],
    "summary": "Determine whether the graph induced by any simple arrangement of great circles on the sphere is vertex 3-colorable.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Coloring great-circle arrangement graphs",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "felsner2000coloring"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/3-colorability-of-arrangements-of-great-circles/",
    "relatedNews": []
  },
  {
    "slug": "3d-minimum-bend-orthogonal-graph-drawings",
    "title": "3D Minimum-Bend Orthogonal Graph Drawings",
    "aliases": [],
    "summary": "Determine whether every graph of maximum degree at most six has a crossing-free three-dimensional orthogonal drawing with at most two bends per edge.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Low-bend orthogonal graph drawings",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "eades2000three"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/3d-minimum-bend-orthogonal-graph-drawings/",
    "relatedNews": []
  },
  {
    "slug": "3sum-hard-problems",
    "title": "3SUM Hard Problems",
    "aliases": [],
    "summary": "The 3SUM decision problem asks whether three integer sets contain elements a, b, and c satisfying a+b=c. The open question is whether 3SUM and the problems to which it reduces admit algorithms with a polynomial saving over quadratic time. Algorithms with logarithmic-factor improvements are known, but such an exponent saving is conjectured to be impossible even in expectation.",
    "field": "Computational complexity and cryptography",
    "subfield": "Problems, reductions and completeness",
    "topic": "Fine-grained complexity of 3SUM",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "chan2018speedups"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/3sum-hard-problems/",
    "relatedNews": []
  },
  {
    "slug": "bpp-vs-p",
    "title": "BPP versus P",
    "aliases": [
      "Does randomness help efficient computation?"
    ],
    "summary": "Determine whether every bounded-error randomized polynomial-time algorithm can be replaced by a deterministic polynomial-time algorithm.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Pseudorandomness and derandomization",
    "topic": "BPP derandomization",
    "standing": {
      "label": "Open; major hardness-versus-randomness results give conditional derandomizations",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "impagliazzo-wigderson"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/bpp-vs-p/",
    "relatedNews": []
  },
  {
    "slug": "chromatic-number-of-the-plane",
    "title": "Chromatic Number of the Plane",
    "aliases": [],
    "summary": "Determine the minimum number of colors needed to color the Euclidean plane so that points at unit distance receive different colors.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Unit-distance coloring of the plane",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "erdos1951colouring"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/chromatic-number-of-the-plane/",
    "relatedNews": []
  },
  {
    "slug": "compatible-triangulations",
    "title": "Compatible Triangulations",
    "aliases": [],
    "summary": "Determine whether any two planar point sets of the same size and with the same number of hull vertices admit combinatorially compatible triangulations.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Compatible point-set triangulations",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "aichholzer2002compatible"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/compatible-triangulations/",
    "relatedNews": []
  },
  {
    "slug": "congruent-partitions-of-polygons",
    "title": "Congruent Partitions of Polygons",
    "aliases": [],
    "summary": "For a polygon and a prescribed number of pieces, find mutually congruent pieces that leave the least uncovered area and characterize when a perfect congruent partition exists.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Optimal congruent polygon partitions",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "elkhechen2008partitioning"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/congruent-partitions-of-polygons/",
    "relatedNews": []
  },
  {
    "slug": "counting-polyominoes",
    "title": "Counting Polyominoes",
    "aliases": [],
    "summary": "Determine the asymptotic growth constant and sharper enumeration bounds for fixed polyominoes of order n.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Generating random combinatorial structures",
    "topic": "Asymptotic enumeration of polyominoes",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "barequet2015improved"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/counting-polyominoes/",
    "relatedNews": []
  },
  {
    "slug": "distances-among-point-sets-in-r-2-and-r-3",
    "title": "Distances among Point Sets in R^2 and R^3",
    "aliases": [],
    "summary": "Determine tight extremal bounds for the number of unit-distance pairs and the minimum number of distinct distances among finite point sets in two and three dimensions.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Unit and distinct distance extremal bounds",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "erdos1946distances"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/distances-among-point-sets-in-r-2-and-r-3/",
    "relatedNews": []
  },
  {
    "slug": "dynamic-planar-nearest-neighbors",
    "title": "Dynamic Planar Nearest Neighbors",
    "aliases": [],
    "summary": "Maintain a dynamic planar point set under insertions and deletions while supporting nearest-neighbor queries, with logarithmic time for every operation.",
    "field": "Design and analysis of algorithms",
    "subfield": "Data structures design and analysis",
    "topic": "Logarithmic dynamic nearest neighbors",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "agarwal1995dynamic"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/dynamic-planar-nearest-neighbors/",
    "relatedNews": []
  },
  {
    "slug": "edge-coloring-geometric-graphs",
    "title": "Edge-Coloring Geometric Graphs",
    "aliases": [],
    "summary": "Determine the minimum number of colors required to color all edges of a complete geometric graph so that crossing edges receive different colors.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Crossing-free edge color classes",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "bose2006partitions"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/edge-coloring-geometric-graphs/",
    "relatedNews": []
  },
  {
    "slug": "edge-unfolding-convex-polyhedra",
    "title": "Edge-Unfolding Convex Polyhedra",
    "aliases": [],
    "summary": "The open problem asks whether every convex polyhedron can be cut along its edges and unfolded into a single simple polygon in the plane without overlap. The answer is conjectured to be affirmative, but no general result is known.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Edge-unfolding convex polyhedra",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "bern2003ununfoldable"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/edge-unfolding-convex-polyhedra/",
    "relatedNews": []
  },
  {
    "slug": "edge-unfolding-polycubes",
    "title": "Edge-Unfolding Polycubes",
    "aliases": [],
    "summary": "Determine whether every genus-zero polycube can be cut along its unit-square edges and unfolded into one nonoverlapping planar piece.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Edge unfoldings of genus-zero polycubes",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "biedl1998unfolding"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/edge-unfolding-polycubes/",
    "relatedNews": []
  },
  {
    "slug": "equiprojective-polyhedra",
    "title": "Equiprojective Polyhedra",
    "aliases": [],
    "summary": "For each integer k, identify or construct all polyhedra whose generic orthogonal projections always have exactly k sides.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Classification of equiprojective polyhedra",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "hasan2008equiprojective"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/equiprojective-polyhedra/",
    "relatedNews": []
  },
  {
    "slug": "euclidean-minimum-spanning-tree",
    "title": "Euclidean Minimum Spanning Tree",
    "aliases": [],
    "summary": "The open problem asks whether the Euclidean minimum spanning tree of n points in R^d can be computed in time close to the (n n) lower bound. The difficulty is most pronounced in large dimension, where the reported upper bounds approach quadratic time.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Euclidean minimum spanning trees",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "callahan1995decomposition"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/euclidean-minimum-spanning-tree/",
    "relatedNews": []
  },
  {
    "slug": "matrix-multiplication-exponent",
    "title": "Exponent of matrix multiplication",
    "aliases": [
      "Is omega equal to 2?"
    ],
    "summary": "Determine whether two dense square matrices over a field can be multiplied using essentially quadratic arithmetic operations.",
    "field": "Computational complexity and cryptography",
    "subfield": "Algebraic complexity theory",
    "topic": "Matrix multiplication exponent",
    "standing": {
      "label": "Open; the best proved exponent is above the information-theoretic lower bound of two",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "williams-multiplication"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/matrix-multiplication-exponent/",
    "relatedNews": [
      "alphaevolve-matrix-multiplication"
    ]
  },
  {
    "slug": "extending-pseudosegment-arrangements-by-subdivision",
    "title": "Extending Pseudosegment Arrangements by Subdivision",
    "aliases": [],
    "summary": "Determine the minimum worst-case number of subdivision vertices needed to extend an arrangement of pseudosegments into a pseudoline arrangement.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Pseudosegment extension complexity",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "chan2000pseudosegments"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/extending-pseudosegment-arrangements-by-subdivision/",
    "relatedNews": []
  },
  {
    "slug": "fair-partitioning-of-convex-polygons",
    "title": "Fair Partitioning of Convex Polygons",
    "aliases": [],
    "summary": "Determine whether every convex polygon can be partitioned into any prescribed number of convex pieces having equal areas and equal perimeters.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Equal-area equal-perimeter convex partitions",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "aronov2010convex"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/fair-partitioning-of-convex-polygons/",
    "relatedNews": []
  },
  {
    "slug": "fast-l-1-difference-a1774140",
    "title": "Fast L_1 Difference",
    "aliases": [],
    "summary": "The problem concerns the update-time complexity of computing the L_1 difference between two vectors specified by data streams. The standard approach described by the source uses projections onto pseudorandom vectors whose entries are drawn from the Cauchy distribution, but sufficient accuracy requires many independent inner products and can make each update costly. The open directions are to obtain faster L_1-difference algorithms through large-frequency or sparse-projection techniques and to prove nontrivial worst-case or amortized lower bounds for stream-update time.",
    "field": "Design and analysis of algorithms",
    "subfield": "Streaming, sublinear and near linear time algorithms",
    "topic": "Fast L_1 difference estimation",
    "standing": {
      "label": "Community submission; not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "sublinear2006fastl1"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/fast-l-1-difference-a1774140/",
    "relatedNews": []
  },
  {
    "slug": "flip-graph-connectivity-in-3d",
    "title": "Flip Graph Connectivity in 3D",
    "aliases": [],
    "summary": "For a finite point set in three-dimensional general position, determine whether all tetrahedralizations are connected by local bistellar flips.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Connectivity of tetrahedralization flips",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "demaine2004small"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/flip-graph-connectivity-in-3d/",
    "relatedNews": []
  },
  {
    "slug": "freeze-tag-optimal-strategies-for-awakening-a-swarm-of-robots",
    "title": "Freeze-Tag: Optimal Strategies for Awakening a Swarm of Robots",
    "aliases": [],
    "summary": "Determine the computational complexity and approximability of awakening sleeping robots in metric and planar geometric spaces as quickly as possible.",
    "field": "Design and analysis of algorithms",
    "subfield": "Approximation algorithms analysis",
    "topic": "Freeze-tag scheduling and complexity",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "arkin2002freeze"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/freeze-tag-optimal-strategies-for-awakening-a-swarm-of-robots/",
    "relatedNews": []
  },
  {
    "slug": "general-unfoldings-of-nonconvex-polyhedra",
    "title": "General Unfoldings of Nonconvex Polyhedra",
    "aliases": [],
    "summary": "Determine whether every closed nonconvex polyhedron has a connected cut set on its surface whose development is a single nonoverlapping planar polygon.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Nonoverlapping nonconvex unfoldings",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "damian2007unfolding"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/general-unfoldings-of-nonconvex-polyhedra/",
    "relatedNews": []
  },
  {
    "slug": "graph-isomorphism-complexity",
    "title": "Graph Isomorphism in polynomial time",
    "aliases": [
      "Complexity of Graph Isomorphism"
    ],
    "summary": "Determine whether graph isomorphism has a deterministic polynomial-time algorithm for arbitrary finite graphs.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Graph isomorphism",
    "standing": {
      "label": "Open; a quasipolynomial-time algorithm is known, but no polynomial-time algorithm for general graphs is known",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "babai-gi"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/graph-isomorphism-complexity/",
    "relatedNews": []
  },
  {
    "slug": "hamiltonian-tetrahedralizations",
    "title": "Hamiltonian Tetrahedralizations",
    "aliases": [],
    "summary": "Determine whether every convex three-dimensional polytope has a tetrahedralization whose dual graph contains a Hamiltonian path.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Hamiltonian duals of tetrahedralizations",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "arkin1996hamiltonian"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/hamiltonian-tetrahedralizations/",
    "relatedNews": []
  },
  {
    "slug": "k-sets",
    "title": "k-sets",
    "aliases": [],
    "summary": "Given a set of points, a k-set is a subset of k points obtained by intersecting the point set with an open halfspace. The open problem is to determine the maximum possible number of k-sets. Even in two dimensions, the known upper and lower bounds remain separated.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Extremal counts of planar k-sets",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "dey1998planarksets"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/k-sets/",
    "relatedNews": []
  },
  {
    "slug": "l-vs-nl",
    "title": "L versus NL",
    "aliases": [
      "Deterministic versus nondeterministic logspace"
    ],
    "summary": "Determine whether nondeterminism adds computational power when a machine is restricted to logarithmic working space.",
    "field": "Computational complexity and cryptography",
    "subfield": "Complexity classes",
    "topic": "Directed reachability",
    "standing": {
      "label": "Open; directed reachability remains the canonical NL-complete problem",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "complexity-zoo-l"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/l-vs-nl/",
    "relatedNews": []
  },
  {
    "slug": "linear-programming-strongly-polynomial",
    "title": "Linear Programming: Strongly Polynomial?",
    "aliases": [],
    "summary": "The open problem is whether linear programming admits a strongly polynomial algorithm. Linear programming is known to be weakly polynomial, meaning polynomial in the bit complexity of the input, and strongly polynomial linear-time algorithms are known when the dimension is fixed.",
    "field": "Design and analysis of algorithms",
    "subfield": "Mathematical optimization",
    "topic": "Strongly polynomial linear programming",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "dyer1984linear"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/linear-programming-strongly-polynomial/",
    "relatedNews": []
  },
  {
    "slug": "linear-volume-3d-grid-drawings-of-planar-graphs",
    "title": "Linear-Volume 3D Grid Drawings of Planar Graphs",
    "aliases": [],
    "summary": "Determine whether every planar graph has a crossing-free straight-line three-dimensional grid drawing of linear volume.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Volume of three-dimensional planar drawings",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "dujmovic2004three"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/linear-volume-3d-grid-drawings-of-planar-graphs/",
    "relatedNews": []
  },
  {
    "slug": "lines-tangent-to-four-unit-balls",
    "title": "Lines Tangent to Four Unit Balls",
    "aliases": [],
    "summary": "Determine the maximum number of lines tangent to four unit balls in three dimensions while avoiding the interiors of all other balls in the set.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Counting unobstructed ball tangents",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "agarwal2005lines"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/lines-tangent-to-four-unit-balls/",
    "relatedNews": []
  },
  {
    "slug": "minimum-euclidean-matching-in-2d",
    "title": "Minimum Euclidean Matching in 2D",
    "aliases": [],
    "summary": "For 2n points in the plane, the cost of a Euclidean matching is the total length of its edges. The open problem is to determine the computational complexity of finding a matching of minimum cost. An exact algorithm with running time O(n^1.5 ^5 n) is reported.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Euclidean minimum-cost matching",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "vaidya1989matching"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/minimum-euclidean-matching-in-2d/",
    "relatedNews": []
  },
  {
    "slug": "minimum-link-path-in-2d",
    "title": "Minimum-Link Path in 2D",
    "aliases": [],
    "summary": "Determine whether a minimum-link path between two points in a planar polygonal domain can be computed in subquadratic time.",
    "field": "Design and analysis of algorithms",
    "subfield": "Algorithm design techniques",
    "topic": "Subquadratic minimum-link paths",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "mitchell1992minimum"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/minimum-link-path-in-2d/",
    "relatedNews": []
  },
  {
    "slug": "monochromatic-triangles",
    "title": "Monochromatic Triangles",
    "aliases": [],
    "summary": "For a prescribed triangle, determine whether the plane admits a three-coloring containing no monochromatic congruent copy of that triangle.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Avoiding monochromatic congruent triangles",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "graham2004euclidean"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/monochromatic-triangles/",
    "relatedNews": []
  },
  {
    "slug": "most-circular-partition-of-a-square",
    "title": "Most Circular Partition of a Square",
    "aliases": [],
    "summary": "Partition a square into finitely many pieces so as to minimize the maximum ratio of circumradius to inradius among the pieces.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Circularity-optimal square partitions",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "damian2003partitioning"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/most-circular-partition-of-a-square/",
    "relatedNews": []
  },
  {
    "slug": "output-sensitive-convex-hull-in-r-d",
    "title": "Output-sensitive Convex Hull in R^d",
    "aliases": [],
    "summary": "For a set of n points in R^d whose convex hull has f faces, determine the optimal output-sensitive running time for constructing the hull.",
    "field": "Design and analysis of algorithms",
    "subfield": "Algorithm design techniques",
    "topic": "Output-sensitive high-dimensional convex hulls",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "chan1996output"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/output-sensitive-convex-hull-in-r-d/",
    "relatedNews": []
  },
  {
    "slug": "p-vs-np",
    "title": "P versus NP",
    "aliases": [
      "P vs NP"
    ],
    "summary": "Determine whether every decision problem whose solutions can be verified in polynomial time can also be solved in polynomial time.",
    "field": "Computational complexity and cryptography",
    "subfield": "Problems, reductions and completeness",
    "topic": "P versus NP",
    "standing": {
      "label": "Open; listed by the Clay Mathematics Institute as a Millennium Prize Problem",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "cook-clay"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/p-vs-np/",
    "relatedNews": []
  },
  {
    "slug": "p-vs-pspace",
    "title": "P versus PSPACE",
    "aliases": [
      "Polynomial time versus polynomial space"
    ],
    "summary": "Determine whether every problem solvable with polynomial working space is also solvable in polynomial time.",
    "field": "Computational complexity and cryptography",
    "subfield": "Complexity classes",
    "topic": "Polynomial space",
    "standing": {
      "label": "Open; equality would collapse all classes between P and PSPACE",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "complexity-zoo-pspace"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/p-vs-pspace/",
    "relatedNews": []
  },
  {
    "slug": "pallet-loading",
    "title": "Pallet Loading",
    "aliases": [],
    "summary": "Determine the computational complexity of deciding how many congruent axis-parallel rectangles, with rotations allowed, fit in a larger rectangle.",
    "field": "Computational complexity and cryptography",
    "subfield": "Problems, reductions and completeness",
    "topic": "Complexity of orthogonal pallet packing",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "tarnowsky1992algorithm"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/pallet-loading/",
    "relatedNews": []
  },
  {
    "slug": "planar-euclidean-maximum-tsp",
    "title": "Planar Euclidean Maximum TSP",
    "aliases": [],
    "summary": "Determine the computational complexity of finding a maximum-length traveling-salesperson tour through points in the Euclidean plane.",
    "field": "Computational complexity and cryptography",
    "subfield": "Problems, reductions and completeness",
    "topic": "Complexity of maximum Euclidean tours",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "barvinok1996maximum"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/planar-euclidean-maximum-tsp/",
    "relatedNews": []
  },
  {
    "slug": "point-location-in-3d-subdivision",
    "title": "Point Location in 3D Subdivision",
    "aliases": [],
    "summary": "The open problem is to construct a linear-space data structure for point location in a three-dimensional subdivision that answers every query in logarithmic time. For a subdivision with n faces, the reported known result uses O(n n) space and answers queries in O( ^2 n) time.",
    "field": "Design and analysis of algorithms",
    "subfield": "Data structures design and analysis",
    "topic": "Point location in 3D subdivisions",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "snoeyink1997pointlocation"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/point-location-in-3d-subdivision/",
    "relatedNews": []
  },
  {
    "slug": "polygonal-curve-simplification",
    "title": "Polygonal Curve Simplification",
    "aliases": [],
    "summary": "Given a polygonal curve and an error tolerance, compute a simplification using the fewest original vertices and ask whether an optimal solution is possible in nearly linear time.",
    "field": "Design and analysis of algorithms",
    "subfield": "Algorithm design techniques",
    "topic": "Near-linear optimal curve simplification",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "agarwal2000efficient"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/polygonal-curve-simplification/",
    "relatedNews": []
  },
  {
    "slug": "polyhedral-surface-approximation",
    "title": "Polyhedral Surface Approximation",
    "aliases": [],
    "summary": "Given a triangulated surface in three dimensions and a tolerance, efficiently construct a simpler polyhedral surface within the prescribed error.",
    "field": "Design and analysis of algorithms",
    "subfield": "Approximation algorithms analysis",
    "topic": "Approximating triangulated surfaces",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "agarwal1998surface"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/polyhedral-surface-approximation/",
    "relatedNews": []
  },
  {
    "slug": "polyhedron-with-regular-pentagon-faces",
    "title": "Polyhedron with Regular Pentagon Faces",
    "aliases": [],
    "summary": "Determine whether every immersed spherical polyhedral surface made entirely of congruent regular pentagons is assembled from solid dodecahedra glued facet to facet.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Classification of pentagonal polyhedral surfaces",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": []
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/polyhedron-with-regular-pentagon-faces/",
    "relatedNews": []
  },
  {
    "slug": "quantum-pcp",
    "title": "Quantum PCP Conjecture",
    "aliases": [
      "qPCP"
    ],
    "summary": "Determine whether approximating the ground-state energy of a local quantum Hamiltonian remains QMA-hard at constant precision.",
    "field": "Computational complexity and cryptography",
    "subfield": "Quantum complexity theory",
    "topic": "Quantum PCP",
    "standing": {
      "label": "Open in its standard constant-gap formulation",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "aharonov-arad-vidick"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/quantum-pcp/",
    "relatedNews": []
  },
  {
    "slug": "queue-number-of-planar-graphs",
    "title": "Queue-Number of Planar Graphs",
    "aliases": [],
    "summary": "Determine whether planar graphs have queue number bounded by an absolute constant.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Queue layouts of planar graphs",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "heath1992comparing"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/queue-number-of-planar-graphs/",
    "relatedNews": []
  },
  {
    "slug": "rectangling-a-rectangle",
    "title": "Rectangling a Rectangle",
    "aliases": [],
    "summary": "Determine whether a rectangle with rational side lengths can be tiled by finitely many rectangles of equal area whose perimeters are all distinct.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Equal-area unequal-perimeter rectangle tilings",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": []
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/rectangling-a-rectangle/",
    "relatedNews": []
  },
  {
    "slug": "reflexivity-of-point-sets",
    "title": "Reflexivity of Point Sets",
    "aliases": [],
    "summary": "Determine the maximum, over planar point sets of size n, of the minimum number of reflex vertices in a polygonalization.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Minimum-reflex polygonalizations",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "ackerman2009reflexivity"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/reflexivity-of-point-sets/",
    "relatedNews": []
  },
  {
    "slug": "rolling-a-die-over-a-labeled-board",
    "title": "Rolling a Die over a Labeled Board",
    "aliases": [],
    "summary": "Determine the computational complexity of rolling a labeled die through every cell of a fully labeled rectangular board exactly once while matching each required top-face label.",
    "field": "Computational complexity and cryptography",
    "subfield": "Problems, reductions and completeness",
    "topic": "Fully labeled die-rolling boards",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "buchin2007rolling"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/rolling-a-die-over-a-labeled-board/",
    "relatedNews": []
  },
  {
    "slug": "simple-linear-time-polygon-triangulation",
    "title": "Simple Linear-Time Polygon Triangulation",
    "aliases": [],
    "summary": "A deterministic linear-time algorithm for triangulating a simple polygon is known, but its construction is intricate. The open problem is to find a deterministic linear-time triangulation algorithm that is significantly simpler. Randomization yields simpler algorithms, including one with linear running time.",
    "field": "Design and analysis of algorithms",
    "subfield": "Algorithm design techniques",
    "topic": "Simple deterministic polygon triangulation",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "amato2000triangulation"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/simple-linear-time-polygon-triangulation/",
    "relatedNews": []
  },
  {
    "slug": "simple-polygonalizations",
    "title": "Simple Polygonalizations",
    "aliases": [],
    "summary": "Given a planar point set, count the simple polygons whose vertex set is exactly the given set, and determine whether this count can be computed in polynomial time.",
    "field": "Design and analysis of algorithms",
    "subfield": "Parameterized complexity and exact algorithms",
    "topic": "Counting simple polygonalizations",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "marx2016optimal"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/simple-polygonalizations/",
    "relatedNews": []
  },
  {
    "slug": "smallest-universal-set-of-points-for-planar-graphs",
    "title": "Smallest Universal Set of Points for Planar Graphs",
    "aliases": [],
    "summary": "Determine whether there are linear-size universal planar point sets on which every planar graph of a given order has a crossing-free straight-line drawing.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Universal planar drawing point sets",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "chrobak1989universal"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/smallest-universal-set-of-points-for-planar-graphs/",
    "relatedNews": []
  },
  {
    "slug": "sorting-x-y-pairwise-sums",
    "title": "Sorting X+Y (Pairwise Sums)",
    "aliases": [],
    "summary": "Given two sets of n real numbers, determine whether all n^2 pairwise sums can be sorted in optimal quadratic time.",
    "field": "Design and analysis of algorithms",
    "subfield": "Algorithm design techniques",
    "topic": "Quadratic-time pairwise-sum sorting",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "fredman1976sort"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/sorting-x-y-pairwise-sums/",
    "relatedNews": []
  },
  {
    "slug": "strong-exponential-time-hypothesis",
    "title": "Strong Exponential Time Hypothesis",
    "aliases": [
      "SETH"
    ],
    "summary": "Determine whether satisfiability for unbounded clause width fundamentally requires time approaching two to the number of variables.",
    "field": "Design and analysis of algorithms",
    "subfield": "Parameterized complexity and exact algorithms",
    "topic": "Fine-grained SAT hardness",
    "standing": {
      "label": "Open; extensively used as a hypothesis for fine-grained conditional lower bounds",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "ipz-eth"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/strong-exponential-time-hypothesis/",
    "relatedNews": []
  },
  {
    "slug": "sum-of-square-roots",
    "title": "Sum of Square Roots",
    "aliases": [],
    "summary": "Determine polynomial separation bounds, and hence efficient exact comparison methods, for differences between sums of square roots of bounded integers.",
    "field": "Computational complexity and cryptography",
    "subfield": "Algebraic complexity theory",
    "topic": "Separation bounds for radical sums",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "burnikel2000comparison"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/sum-of-square-roots/",
    "relatedNews": []
  },
  {
    "slug": "np-circuit-lower-bounds",
    "title": "Superpolynomial circuit lower bounds for NP",
    "aliases": [
      "NP not in P/poly"
    ],
    "summary": "Prove that some language in NP requires Boolean circuits of superpolynomial size, equivalently that NP is not contained in P/poly.",
    "field": "Computational complexity and cryptography",
    "subfield": "Circuit complexity",
    "topic": "NP circuit lower bounds",
    "standing": {
      "label": "Open for unrestricted Boolean circuits; strong lower bounds are known for restricted circuit classes",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "razborov-rudich"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/np-circuit-lower-bounds/",
    "relatedNews": []
  },
  {
    "slug": "surface-reconstruction",
    "title": "Surface Reconstruction",
    "aliases": [],
    "summary": "Reconstruct a surface from a sufficiently dense point sample with a guarantee that the output is homeomorphic to the sampled surface, including surfaces with sharp edges and corners.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Surface reconstruction with singularities",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "amenta2000simple"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/surface-reconstruction/",
    "relatedNews": []
  },
  {
    "slug": "the-number-of-pointed-pseudotriangulations",
    "title": "The Number of Pointed Pseudotriangulations",
    "aliases": [],
    "summary": "Determine whether every planar point set has at least as many pointed pseudotriangulations as triangulations, with equality only in convex position.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Counting pointed pseudotriangulations",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "aichholzer2002pseudotriangulations"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/the-number-of-pointed-pseudotriangulations/",
    "relatedNews": []
  },
  {
    "slug": "thrackles",
    "title": "Thrackles",
    "aliases": [],
    "summary": "Determine whether every drawing of a graph in which each pair of edges meets exactly once has at most as many edges as vertices.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Conway's thrackle bound",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "cairns2000bounds"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/thrackles/",
    "relatedNews": []
  },
  {
    "slug": "transforming-polygons-via-vertex-centroid-moves",
    "title": "Transforming Polygons via Vertex-Centroid Moves",
    "aliases": [],
    "summary": "Determine whether every polygon can be transformed into a regular polygon by finitely many moves that send one vertex along its line to the centroid of the remaining vertices.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Centroid-guided polygon transformations",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": []
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/transforming-polygons-via-vertex-centroid-moves/",
    "relatedNews": []
  },
  {
    "slug": "trapping-light-rays-with-segment-mirrors",
    "title": "Trapping Light Rays with Segment Mirrors",
    "aliases": [],
    "summary": "Determine whether a finite collection of pairwise disjoint planar segment mirrors can trap every ray emitted from a source point disjoint from the mirrors.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Light trapping by disjoint mirrors",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "orourke2001narrowing"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/trapping-light-rays-with-segment-mirrors/",
    "relatedNews": []
  },
  {
    "slug": "traveling-salesman-problem-in-solid-grid-graphs",
    "title": "Traveling Salesman Problem in Solid Grid Graphs",
    "aliases": [],
    "summary": "Determine the complexity of finding a shortest traveling-salesperson tour in a solid grid graph.",
    "field": "Computational complexity and cryptography",
    "subfield": "Problems, reductions and completeness",
    "topic": "Shortest tours in solid grid graphs",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "itai1982hamilton"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/traveling-salesman-problem-in-solid-grid-graphs/",
    "relatedNews": []
  },
  {
    "slug": "union-of-fat-objects-in-3d",
    "title": "Union of Fat Objects in 3D",
    "aliases": [],
    "summary": "An object in three-dimensional space is called fat when the ratio of its circumradius to its inradius is bounded. The open problem is to determine the combinatorial complexity of the union of n such objects. This complexity is conjectured to be nearly quadratic in n.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Unions of three-dimensional fat objects",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "agarwal1999pipes"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/union-of-fat-objects-in-3d/",
    "relatedNews": []
  },
  {
    "slug": "unique-games-conjecture",
    "title": "Unique Games Conjecture",
    "aliases": [
      "UGC"
    ],
    "summary": "Determine whether it is NP-hard to distinguish almost satisfiable unique constraint systems from instances in which only a small fraction can be satisfied.",
    "field": "Design and analysis of algorithms",
    "subfield": "Approximation algorithms analysis",
    "topic": "Unique Games hardness",
    "standing": {
      "label": "Open; the conjecture remains a central hypothesis in hardness of approximation",
      "asOf": "2026-08-05",
      "sourceReferenceIds": [
        "khot-ugc"
      ]
    },
    "updatedAt": "2026-08-05",
    "packVersion": 4,
    "url": "/problems/unique-games-conjecture/",
    "relatedNews": []
  },
  {
    "slug": "vertex-pi-floodlights",
    "title": "Vertex pi-Floodlights",
    "aliases": [],
    "summary": "Determine the worst-case number of inward-facing floodlights of aperture , placed at polygon vertices, that suffice to illuminate every simple polygon.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Vertex floodlight illumination",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "speckmann2001vertex"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/vertex-pi-floodlights/",
    "relatedNews": []
  },
  {
    "slug": "vertex-unfolding-polyhedra",
    "title": "Vertex-Unfolding Polyhedra",
    "aliases": [],
    "summary": "Determine whether every closed polyhedron can be cut along edges into a connected, nonoverlapping planar unfolding whose faces may remain joined only at vertices.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Vertex-connected polyhedral unfoldings",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "demaine2002vertex"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/vertex-unfolding-polyhedra/",
    "relatedNews": []
  },
  {
    "slug": "vertical-decompositions-in-r-d",
    "title": "Vertical Decompositions in R^d",
    "aliases": [],
    "summary": "Determine the worst-case combinatorial complexity of the vertical decomposition of an arrangement of n constant-complexity surfaces in R^d for d 5.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Complexity of vertical decompositions",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "agarwal2000arrangements"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/vertical-decompositions-in-r-d/",
    "relatedNews": []
  },
  {
    "slug": "visibility-graph-recognition",
    "title": "Visibility Graph Recognition",
    "aliases": [],
    "summary": "Given a graph and a specified Hamiltonian cycle, determine whether they are respectively the visibility graph and boundary cycle of a simple polygon.",
    "field": "Computational complexity and cryptography",
    "subfield": "Problems, reductions and completeness",
    "topic": "Simple-polygon visibility graph recognition",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "orourke1993visibility"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/visibility-graph-recognition/",
    "relatedNews": []
  },
  {
    "slug": "volume-maximizing-convex-shape",
    "title": "Volume Maximizing Convex Shape",
    "aliases": [],
    "summary": "For a unit-area convex planar region, choose a perimeter-halving fold and maximize the volume enclosed after identifying the two boundary chains.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Volume-maximizing perimeter-halving folds",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "alexander2003folding"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/volume-maximizing-convex-shape/",
    "relatedNews": []
  },
  {
    "slug": "voronoi-diagram-of-lines-in-3d",
    "title": "Voronoi Diagram of Lines in 3D",
    "aliases": [],
    "summary": "For a set of lines or line segments in three-dimensional space, the open question is to determine the combinatorial complexity of its Voronoi diagram. In the Euclidean case, the known lower bound is quadratic in the number of input objects, while the upper bound is essentially cubic. The complexity is conjectured to be nearly quadratic.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Three-dimensional line Voronoi diagrams",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "agarwal2000pipes"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/voronoi-diagram-of-lines-in-3d/",
    "relatedNews": []
  },
  {
    "slug": "yao-yao-graph-a-spanner",
    "title": "Yao-Yao Graph a Spanner?",
    "aliases": [],
    "summary": "Determine whether the Yao-Yao geometric graph has bounded stretch for a fixed number of cones.",
    "field": "Design and analysis of algorithms",
    "subfield": "Graph algorithms analysis",
    "topic": "Stretch of bounded-degree Yao-Yao graphs",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "wang2002distributed"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/yao-yao-graph-a-spanner/",
    "relatedNews": []
  },
  {
    "slug": "zipper-unfoldings-of-convex-polyhedra",
    "title": "Zipper Unfoldings of Convex Polyhedra",
    "aliases": [],
    "summary": "Determine whether every convex polyhedron can be cut open along a single vertex-spanning path and unfolded without overlap.",
    "field": "Randomness, geometry and discrete structures",
    "subfield": "Computational geometry",
    "topic": "Single-path convex polyhedron unfoldings",
    "standing": {
      "label": "Open problem record; imported from a submitted source and not independently verified",
      "asOf": "2026-08-07",
      "sourceReferenceIds": [
        "demaine2010zipper"
      ]
    },
    "updatedAt": "2026-08-07",
    "packVersion": 1,
    "url": "/problems/zipper-unfoldings-of-convex-polyhedra/",
    "relatedNews": []
  }
]