{
  "cases": [
    {
      "case": "parallel-unit-flow",
      "proposal": {
        "status": "feasible_vector_checked",
        "vector": [
          1,
          0
        ],
        "source_arc_order": [
          "direct-one",
          "direct-two"
        ],
        "shifted_solver_objective": 0,
        "original_objective": 0,
        "solver": {
          "library": "NetworkX",
          "version": "3.4.2",
          "algorithm": "network_simplex",
          "graph": "MultiDiGraph",
          "demand_convention": "NetworkX demand=-source shifted balance"
        },
        "optimality_independent_certificate": "not_computed",
        "uniform_sampling": "not_claimed",
        "interpretation": "An existing solver proposes one vector. Exact integer bounds/balances are checked separately. It does not count or sample all feasible flows, and a result from an optimization algorithm is not presented as an independently certified optimum."
      },
      "reference_exact_count": 2
    },
    {
      "case": "two-route-unit-flow",
      "proposal": {
        "status": "feasible_vector_checked",
        "vector": [
          0,
          0,
          1
        ],
        "source_arc_order": [
          "via-a",
          "a-to-t",
          "direct"
        ],
        "shifted_solver_objective": 0,
        "original_objective": 0,
        "solver": {
          "library": "NetworkX",
          "version": "3.4.2",
          "algorithm": "network_simplex",
          "graph": "MultiDiGraph",
          "demand_convention": "NetworkX demand=-source shifted balance"
        },
        "optimality_independent_certificate": "not_computed",
        "uniform_sampling": "not_claimed",
        "interpretation": "An existing solver proposes one vector. Exact integer bounds/balances are checked separately. It does not count or sample all feasible flows, and a result from an optimization algorithm is not presented as an independently certified optimum."
      },
      "reference_exact_count": 2
    },
    {
      "case": "signed-loop-flow",
      "proposal": {
        "status": "feasible_vector_checked",
        "vector": [
          -2
        ],
        "source_arc_order": [
          "loop"
        ],
        "shifted_solver_objective": 0,
        "original_objective": 0,
        "solver": {
          "library": "NetworkX",
          "version": "3.4.2",
          "algorithm": "network_simplex",
          "graph": "MultiDiGraph",
          "demand_convention": "NetworkX demand=-source shifted balance"
        },
        "optimality_independent_certificate": "not_computed",
        "uniform_sampling": "not_claimed",
        "interpretation": "An existing solver proposes one vector. Exact integer bounds/balances are checked separately. It does not count or sample all feasible flows, and a result from an optimization algorithm is not presented as an independently certified optimum."
      },
      "reference_exact_count": 5
    },
    {
      "case": "binary-capacity-flow",
      "proposal": {
        "status": "feasible_vector_checked",
        "vector": [
          1
        ],
        "source_arc_order": [
          "large-binary-capacity"
        ],
        "shifted_solver_objective": 0,
        "original_objective": 0,
        "solver": {
          "library": "NetworkX",
          "version": "3.4.2",
          "algorithm": "network_simplex",
          "graph": "MultiDiGraph",
          "demand_convention": "NetworkX demand=-source shifted balance"
        },
        "optimality_independent_certificate": "not_computed",
        "uniform_sampling": "not_claimed",
        "interpretation": "An existing solver proposes one vector. Exact integer bounds/balances are checked separately. It does not count or sample all feasible flows, and a result from an optimization algorithm is not presented as an independently certified optimum."
      },
      "reference_exact_count": null
    }
  ],
  "public_case": {
    "provenance": {
      "source": "https://developers.google.com/optimization/flow/mincostflow",
      "source_example": "Public 5-node, 9-arc model in official OR-Tools guide",
      "original_costs": {
        "arc-0": 4,
        "arc-1": 4,
        "arc-2": 2,
        "arc-3": 2,
        "arc-4": 6,
        "arc-5": 1,
        "arc-6": 3,
        "arc-7": 2,
        "arc-8": 3
      },
      "source_count_model": "All capacity/balance-feasible integer arc vectors; costs do not restrict this set",
      "networkx_optimization_model": "Original costs used by existing solver; no independent optimality certificate computed",
      "real_customer_dataset": false,
      "ortools_executed": false
    },
    "proposal": {
      "status": "feasible_vector_checked",
      "vector": [
        12,
        8,
        8,
        4,
        0,
        12,
        4,
        11,
        0
      ],
      "source_arc_order": [
        "arc-0",
        "arc-1",
        "arc-2",
        "arc-3",
        "arc-4",
        "arc-5",
        "arc-6",
        "arc-7",
        "arc-8"
      ],
      "shifted_solver_objective": 150,
      "original_objective": 150,
      "solver": {
        "library": "NetworkX",
        "version": "3.4.2",
        "algorithm": "network_simplex",
        "graph": "MultiDiGraph",
        "demand_convention": "NetworkX demand=-source shifted balance"
      },
      "optimality_independent_certificate": "not_computed",
      "uniform_sampling": "not_claimed",
      "interpretation": "An existing solver proposes one vector. Exact integer bounds/balances are checked separately. It does not count or sample all feasible flows, and a result from an optimization algorithm is not presented as an independently certified optimum."
    },
    "reference": {
      "status": "reduction_complete_count_unknown",
      "exact_count": null,
      "feasible": null,
      "flows": null,
      "arc_diagnostics": null,
      "arc_order": [
        "arc-0",
        "arc-1",
        "arc-2",
        "arc-3",
        "arc-4",
        "arc-5",
        "arc-6",
        "arc-7",
        "arc-8"
      ],
      "reduction": {
        "table": {
          "rows": [
            263,
            243,
            243,
            243,
            243,
            243,
            243,
            243,
            243,
            243,
            243,
            243,
            243,
            243
          ],
          "columns": [
            243,
            243,
            243,
            248,
            258,
            243,
            243,
            243,
            243,
            243,
            243,
            243,
            243,
            243
          ],
          "cell_bounds": [
            [
              243,
              0,
              0,
              0,
              0,
              15,
              8,
              0,
              0,
              0,
              0,
              0,
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            ],
            [
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              243,
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              20,
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              10,
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            ],
            [
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              15,
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            [
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            [
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            [
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            [
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            ],
            [
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              20,
              0,
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              0,
              243,
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            ],
            [
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              0,
              4,
              0,
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              0,
              243,
              0,
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              0,
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              0
            ],
            [
              0,
              0,
              0,
              0,
              10,
              0,
              0,
              0,
              0,
              243,
              0,
              0,
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            ],
            [
              0,
              0,
              0,
              15,
              0,
              0,
              0,
              0,
              0,
              0,
              243,
              0,
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            ],
            [
              0,
              0,
              0,
              0,
              4,
              0,
              0,
              0,
              0,
              0,
              0,
              243,
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            ],
            [
              0,
              0,
              0,
              0,
              20,
              0,
              0,
              0,
              0,
              0,
              0,
              0,
              243,
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            ],
            [
              0,
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              0,
              0,
              0,
              0,
              0,
              0,
              0,
              0,
              0,
              0,
              243
            ]
          ]
        },
        "K": 243,
        "dimension": 14,
        "total": 3422,
        "index_order": [
          {
            "kind": "original_vertex",
            "id": "0"
          },
          {
            "kind": "original_vertex",
            "id": "1"
          },
          {
            "kind": "original_vertex",
            "id": "2"
          },
          {
            "kind": "original_vertex",
            "id": "3"
          },
          {
            "kind": "original_vertex",
            "id": "4"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-0"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-1"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-2"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-3"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-4"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-5"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-6"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-7"
          },
          {
            "kind": "private_arc_vertex",
            "id": "arc-8"
          }
        ],
        "balance_convention": "Outflow minus inflow",
        "lower_bound_shifted_balances": {
          "0": 20,
          "1": 0,
          "2": 0,
          "3": -5,
          "4": -15
        },
        "preserves": "One table for every original integer arc-value vector, including distinct parallel arcs and loops"
      },
      "limits": {
        "vertices": 16,
        "arcs": 16,
        "magnitude_bits": 256,
        "exact_table_dimension": 6,
        "exact_table_total": 128,
        "diagnostics": 2000,
        "state_budget": 50000,
        "count_work_budget": 1000000,
        "enum_work_budget": 4000000
      },
      "source_connection": {
        "family": "115",
        "source_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
        "paper_section": "An FPRAS for Cell-Bounded Contingency Tables, bounded integral flows",
        "source_FPRAS_or_approximate_sampler_implemented": false,
        "selected_comparator_contains_separate_flow_statement": false
      },
      "source_proof_verification": "not_run",
      "formal_program_verification": "not_run",
      "interpretation": "Finite reduction/count diagnostics. A state is an arc vector, not a path decomposition. Uniform feasible flows describe a chosen scenario law, not real failure probabilities, route optimality or a multicommodity model. Diagnostic/budget exhaustion remains unknown, never zero.",
      "reason": "Valid reduction exceeds the finite table-reference domain; count and feasibility unknown"
    }
  }
}