POST /api/v1/integer/solve — Pure Integer Programming

The /api/v1/integer/solve endpoint solves Pure Integer Programming (IP) problems where every decision variable must take a non-negative integer value. The solver uses Branch & Bound with LP relaxation at each node, branching by applying floor/ceil bounds to fractional variables until an integer-feasible optimal solution is confirmed or the problem is shown to be infeasible.

Request

POST /api/v1/integer/solve

Content-Type: application/json

objective

list[float]

required

Coefficients of the objective function, one per decision variable. Must contain between 2 and 5 elements. Variables are named x1, x2, …, x5 in order.Example: [5, 8] represents 5x1 + 8x2.

goal

string

required

Optimisation direction. Accepted values: "max" or "min".

constraints

list[Constraint]

required

At least one linear constraint. Every constraint must have the same number of coefficients as objective.

Show Constraint object

coefficients

list[float]

required

Left-hand-side coefficients, one per decision variable (2–5 elements, must match the length of objective).

inequality

string

required

Relationship operator. One of "<=", ">=", or "=".

rhs

float

required

Right-hand-side constant of the constraint.

Response

A successful call returns an IntegerResponse object.

status

string

Outcome of the solve. One of:

"optimal" — a best integer solution was found.
"infeasible" — no feasible integer assignment satisfies all constraints.
"limit" — the node limit was reached before proving optimality.

objective_value

float | null

Optimal objective value at the best integer solution found. null when status is "infeasible".

variables

object | null

Map of variable names to their optimal integer values, e.g. {"x1": 3, "x2": 2}. null when status is "infeasible".

nodes_explored

integer

Total number of Branch & Bound nodes evaluated during the search.

nodes

list[IntegerNode]

Complete Branch & Bound exploration tree, one entry per node.

Show IntegerNode object

node_id

integer

Unique identifier of this B&B node (root is 0).

parent_id

integer | null

node_id of the parent node. null for the root node.

depth

integer

Depth of this node in the B&B tree (root is depth 0).

lower_bounds

object

Active lower bounds on variables at this node, e.g. {"x1": 3.0}. Inherits and extends the parent’s bounds.

upper_bounds

object

Active upper bounds on variables at this node, e.g. {"x1": 3.0}. Inherits and extends the parent’s bounds.

lp_value

float | null

Optimal LP relaxation objective value at this node. null if the LP was infeasible.

lp_vars

object | null

Map of variable names to their LP relaxation values at this node. null if the LP was infeasible.

status

string

Node processing result. Common values: "branched", "pruned_bound", "pruned_infeasible", "integer_feasible".

branched_on

string | null

Name of the variable selected for branching at this node (e.g. "x1"). null for leaf nodes.

branch_direction

string | null

Direction of the bound applied when creating this node: "floor" (upper bound set to ⌊value⌋) or "ceil" (lower bound set to ⌈value⌉). null for the root node.

edge_label

string | null

Human-readable label on the tree edge from the parent to this node (e.g. "x1 ≤ 3"). null for the root node.

message

string

Human-readable status message describing the result.

Example

Maximize 5x1 + 8x2 subject to:

x1 + x2 ≤ 6
5x1 + 9x2 ≤ 45

curl -X POST https://api.example.com/api/v1/integer/solve \
  -H "Content-Type: application/json" \
  -d '{
    "objective": [5, 8],
    "goal": "max",
    "constraints": [
      {"coefficients": [1, 1], "inequality": "<=", "rhs": 6},
      {"coefficients": [5, 9], "inequality": "<=", "rhs": 45}
    ]
  }'

{
  "status": "optimal",
  "objective_value": 41.0,
  "variables": {
    "x1": 1,
    "x2": 4
  },
  "nodes_explored": 5,
  "nodes": [
    {
      "node_id": 0,
      "parent_id": null,
      "depth": 0,
      "lower_bounds": {"x1": 0.0, "x2": 0.0},
      "upper_bounds": {},
      "lp_value": 41.25,
      "lp_vars": {"x1": 1.25, "x2": 4.375},
      "status": "branched",
      "branched_on": "x2",
      "branch_direction": null,
      "edge_label": null
    },
    {
      "node_id": 1,
      "parent_id": 0,
      "depth": 1,
      "lower_bounds": {"x1": 0.0, "x2": 0.0},
      "upper_bounds": {"x2": 4.0},
      "lp_value": 41.0,
      "lp_vars": {"x1": 1.0, "x2": 4.0},
      "status": "integer_feasible",
      "branched_on": null,
      "branch_direction": "floor",
      "edge_label": "x2 ≤ 4"
    },
    {
      "node_id": 2,
      "parent_id": 0,
      "depth": 1,
      "lower_bounds": {"x1": 0.0, "x2": 5.0},
      "upper_bounds": {},
      "lp_value": 40.0,
      "lp_vars": {"x1": 0.0, "x2": 5.0},
      "status": "pruned_bound",
      "branched_on": null,
      "branch_direction": "ceil",
      "edge_label": "x2 ≥ 5"
    }
  ],
  "message": "Optimal integer solution found"
}

Unlike the Binary solver, Pure Integer Programming allows each variable to take any non-negative integer value. The solver applies tighter lower_bounds and upper_bounds progressively as it branches deeper into the tree.

All constraint coefficients arrays must have the same length as objective. Mismatched lengths return HTTP 422 Unprocessable Entity.

Health Check

GET /api/v1/integer/health Returns a liveness confirmation for the Pure Integer Programming service.

curl https://api.example.com/api/v1/integer/health

{
  "status": "ok",
  "service": "integer-optimizer-api"
}

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