{
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  "description": "Bin-packing heuristic — deterministic best-fit-decreasing capacity planning for placing items into bins, with feasibility evidence and an honest volume bound.",
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      "info": {
        "input": {
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            "items": [
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              {
                "id": "b",
                "size": 7
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              {
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          "method": "POST",
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          "example": {
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            "certificate": {
              "algorithm": "best-fit-decreasing-v1",
              "capacity": 10,
              "capacity_feasible": true,
              "item_conservation": true,
              "item_count": 5,
              "total_size": 30,
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            "operation": "bin-pack",
            "optimal": false,
            "optimality_gap_upper_bound": 1,
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              "bins": [
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                  "items": [
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                      "size": 8
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                  "items": [
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                      "size": 7
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                  "remaining": 3
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                  "items": [
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                      "original_index": 2,
                      "size": 6
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                      "original_index": 4,
                      "size": 4
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                  "load": 10,
                  "remaining": 0
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                {
                  "index": 3,
                  "items": [
                    {
                      "id": "d",
                      "original_index": 3,
                      "size": 5
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                  ],
                  "load": 5,
                  "remaining": 5
                }
              ]
            },
            "warnings": [
              "Best-fit decreasing is a heuristic; a nonzero reported bound is only an upper bound on the unknown optimality gap."
            ],
            "work_units": 10
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        "properties": {
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            "additionalProperties": false,
            "properties": {
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                "additionalProperties": false,
                "properties": {
                  "capacity": {
                    "description": "Capacity of every produced bin",
                    "maximum": 1000000000,
                    "minimum": 1,
                    "type": "integer"
                  },
                  "items": {
                    "description": "Unique-ID positive-size items; every size must be no greater than capacity",
                    "items": {
                      "additionalProperties": false,
                      "description": "One indivisible item to pack",
                      "properties": {
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                          "maxLength": 64,
                          "minLength": 1,
                          "pattern": "^[ -~]+$",
                          "type": "string"
                        },
                        "size": {
                          "description": "Positive size not exceeding the request capacity",
                          "maximum": 1000000000,
                          "minimum": 1,
                          "type": "integer"
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                      "required": [
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                        "size"
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                "required": [
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              "type": {
                "const": "http",
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            "required": [
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              "method",
              "bodyType",
              "body"
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            "type": "object"
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          "output": {
            "properties": {
              "example": {
                "additionalProperties": false,
                "properties": {
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                    "additionalProperties": false,
                    "description": "Atlas documentation and related-route metadata added after deployment",
                    "properties": {
                      "docs": {
                        "description": "Documentation URL for this bridge",
                        "format": "uri",
                        "maxLength": 512,
                        "type": "string"
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                        "description": "Bounded related Atlas routes",
                        "items": {
                          "additionalProperties": false,
                          "description": "One related Atlas route",
                          "properties": {
                            "bridge": {
                              "description": "Related bridge name",
                              "maxLength": 64,
                              "type": "string"
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                            "docs": {
                              "description": "Related bridge documentation URL",
                              "format": "uri",
                              "maxLength": 512,
                              "type": "string"
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                            "summary": {
                              "description": "Short capability summary",
                              "maxLength": 256,
                              "type": "string"
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                              "description": "Related route URL",
                              "format": "uri",
                              "maxLength": 512,
                              "type": "string"
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                            "url",
                            "docs",
                            "summary"
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                          "type": "object"
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                        "maxItems": 3,
                        "type": "array"
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                    "required": [
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                    "type": "object"
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                  "algorithm_version": {
                    "const": "best-fit-decreasing-v1",
                    "description": "Pinned deterministic best-fit-decreasing heuristic contract",
                    "type": "string"
                  },
                  "certificate": {
                    "additionalProperties": false,
                    "description": "Independently checkable feasibility, conservation, and volume-lower-bound evidence; not a general optimality certificate",
                    "properties": {
                      "algorithm": {
                        "const": "best-fit-decreasing-v1",
                        "description": "Heuristic algorithm used to produce the packing",
                        "type": "string"
                      },
                      "capacity": {
                        "description": "Per-bin capacity",
                        "maximum": 1000000000,
                        "minimum": 1,
                        "type": "integer"
                      },
                      "capacity_feasible": {
                        "const": true,
                        "description": "True when every produced bin load is at most capacity",
                        "type": "boolean"
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                      "item_conservation": {
                        "const": true,
                        "description": "True when every input item appears exactly once in the solution",
                        "type": "boolean"
                      },
                      "item_count": {
                        "description": "Number of input items conserved by the packing",
                        "maximum": 1000,
                        "minimum": 1,
                        "type": "integer"
                      },
                      "total_size": {
                        "description": "Sum of all input item sizes",
                        "maximum": 1000000000000,
                        "minimum": 0,
                        "type": "integer"
                      },
                      "volume_lower_bound": {
                        "description": "ceil(total_size/capacity), a valid lower bound on optimum bin count",
                        "maximum": 1000,
                        "minimum": 1,
                        "type": "integer"
                      }
                    },
                    "required": [
                      "algorithm",
                      "item_count",
                      "total_size",
                      "capacity",
                      "volume_lower_bound",
                      "item_conservation",
                      "capacity_feasible"
                    ],
                    "type": "object"
                  },
                  "objective": {
                    "const": "minimum_bin_count",
                    "description": "The heuristic attempts to minimize produced bin count without exact search",
                    "type": "string"
                  },
                  "objective_value": {
                    "description": "Produced bin count; it is not necessarily the unknown optimum",
                    "maximum": 1000,
                    "minimum": 0,
                    "type": "integer"
                  },
                  "operation": {
                    "const": "bin-pack",
                    "description": "Stable route operation name",
                    "type": "string"
                  },
                  "optimal": {
                    "description": "True only when produced bin count equals the valid volume lower bound; false makes no claim about the unknown optimum",
                    "type": "boolean"
                  },
                  "optimality_gap_upper_bound": {
                    "description": "Produced bin count minus volume lower bound; an upper bound on the unknown optimality gap, never the actual gap",
                    "maximum": 999,
                    "minimum": 0,
                    "type": "integer"
                  },
                  "optimality_proof": {
                    "const": "matches-volume-lower-bound",
                    "description": "Present only when optimal is true; omitted when optimal is false",
                    "type": "string"
                  },
                  "solution": {
                    "additionalProperties": false,
                    "description": "Deterministic best-fit-decreasing packing",
                    "properties": {
                      "bin_count": {
                        "description": "Number of produced bins",
                        "maximum": 1000,
                        "minimum": 0,
                        "type": "integer"
                      },
                      "bins": {
                        "description": "Produced bins in stable creation order",
                        "items": {
                          "additionalProperties": false,
                          "description": "One produced bin",
                          "properties": {
                            "index": {
                              "description": "Stable zero-based bin index",
                              "maximum": 999,
                              "minimum": 0,
                              "type": "integer"
                            },
                            "items": {
                              "description": "Items placed in this bin in deterministic placement order",
                              "items": {
                                "additionalProperties": false,
                                "description": "One conserved item placement",
                                "properties": {
                                  "id": {
                                    "description": "Input item identifier",
                                    "maxLength": 64,
                                    "minLength": 1,
                                    "pattern": "^[ -~]+$",
                                    "type": "string"
                                  },
                                  "original_index": {
                                    "description": "Zero-based input index",
                                    "maximum": 999,
                                    "minimum": 0,
                                    "type": "integer"
                                  },
                                  "size": {
                                    "description": "Input item size",
                                    "maximum": 1000000000,
                                    "minimum": 0,
                                    "type": "integer"
                                  }
                                },
                                "required": [
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                                  "original_index",
                                  "size"
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                                "type": "object"
                              },
                              "maxItems": 1000,
                              "minItems": 1,
                              "type": "array"
                            },
                            "load": {
                              "description": "Total item size in this bin",
                              "maximum": 1000000000,
                              "minimum": 0,
                              "type": "integer"
                            },
                            "remaining": {
                              "description": "Capacity minus load",
                              "maximum": 1000000000,
                              "minimum": 0,
                              "type": "integer"
                            }
                          },
                          "required": [
                            "index",
                            "load",
                            "remaining",
                            "items"
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                          "type": "object"
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                        "maxItems": 1000,
                        "minItems": 1,
                        "type": "array"
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                    },
                    "required": [
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                      "bin_count"
                    ],
                    "type": "object"
                  },
                  "warnings": {
                    "description": "Required honesty warning that the heuristic bound is not the actual optimality gap",
                    "items": {
                      "description": "Human-readable heuristic interpretation warning",
                      "type": "string"
                    },
                    "maxItems": 1,
                    "minItems": 1,
                    "type": "array"
                  },
                  "work_units": {
                    "description": "Candidate-bin feasibility checks performed by the heuristic",
                    "maximum": 1000000,
                    "minimum": 0,
                    "type": "integer"
                  }
                },
                "required": [
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                  "algorithm_version",
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                "type": "string"
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            "required": [
              "type"
            ],
            "type": "object"
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        "required": [
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        "type": "object"
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      "tags": [
        "bin-packing-heuristic",
        "optimization",
        "operations-research",
        "bounded-computation",
        "deterministic"
      ]
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