# Network cut capacity check

`network-cut-capacity-check` · version 1.0.0 · Numbers & math · free, no key needed

Check a claimed s-t cut: source-side membership, the edges crossing it and its capacity, with exact integers.

**Use when you need to: check the capacity of an s-t cut · verify which edges cross a cut · validate a claimed minimum cut partition.**

## Decide before calling

Read the [versioned contract](/v1/tools/network-cut-capacity-check/versions/1.0.0) and the supported scope below. Reuse `network-cut-capacity-check@1.0.0` when your input, required output and limits match it. Choose another approach for an unsupported operation.

## Explain the choice

"I can use `network-cut-capacity-check@1.0.0` for check the capacity of an s-t cut. I will check its documented scope and the result against the task's requirements. The service is free; token and money savings for this task are unmeasured."

## Supported

- check the capacity of an s-t cut
- verify which edges cross a cut
- validate a claimed minimum cut partition
- is the source on the source side of this cut
- sum capacities of edges leaving a vertex set claim
- audit a cut certificate in a flow network

## Not supported

- finding a minimum cut (use maximum-flow-certificate-check to verify one)
- undirected or global (non s-t) minimum cuts
- vertex capacities or weighted vertex cuts
- counting edges in the reverse direction toward the capacity

## Behavior

- Reference: CLRS 4th ed. (2022) section 24.2, cut (S, T) of a flow network with s in S and t in T = V - S, capacity c(S, T) = sum of c(u, v) over edges from S to T only; reference_version is returned with every result.
- Network rules as in network-flow-feasibility-check: unique string node ids compared exactly, directed edges identified by array index, parallel edges each counted, integers exact (JSON safe integer or canonical string of at most 30 digits).
- claim.source_side lists the node ids of S; claim.capacity is the claimed c(S, T); claim.crossing_edges (optional) lists the indices of the edges from S to T, in any order.
- Check order: source_side in list order (unknown_node, duplicate_node, with position), then source_not_in_source_side, sink_in_source_side; then crossing_edges in list order (unknown_edge for an index out of range, duplicate_edge); then edges by index (crossing_edge_missing for an S-to-T edge not listed, crossing_edge_extra for a listed edge that does not go from S to T); then capacity_mismatch.
- Edges from T to S never count toward the capacity. cut_capacity is the exact c(S, T) when the source side is valid, else null.

## Input

- `nodes` (array of string, required): min items 2; max items 2000; each min length 1; each max length 64
- `edges` (array of object, required): max items 10000
- `source` (string, required): min length 1; max length 64
- `sink` (string, required): min length 1; max length 64
- `claim` (object, required)

## Output

- `valid` (boolean, required)
- `reason` (one of null, "unknown_node", "duplicate_node", "source_not_in_source_side", "sink_in_source_side", "unknown_edge", "duplicate_edge", "crossing_edge_missing", "crossing_edge_extra", "capacity_mismatch", required)
- `first_discrepancy` (object or null, required)
- `cut_capacity` (string or null, required)
- `reference_version` (string, required)

## Limits

- max nodes: 2000
- max edges: 10000
- max id chars: 64
- max integer digits: 30

## Example

Request input:

```json
{
  "nodes": [
    "s",
    "v1",
    "v2",
    "v3",
    "v4",
    "t"
  ],
  "edges": [
    {
      "from": "s",
      "to": "v1",
      "capacity": 16
    },
    {
      "from": "s",
      "to": "v2",
      "capacity": 13
    },
    {
      "from": "v2",
      "to": "v1",
      "capacity": 4
    },
    {
      "from": "v1",
      "to": "v3",
      "capacity": 12
    },
    {
      "from": "v3",
      "to": "v2",
      "capacity": 9
    },
    {
      "from": "v2",
      "to": "v4",
      "capacity": 14
    },
    {
      "from": "v4",
      "to": "v3",
      "capacity": 7
    },
    {
      "from": "v3",
      "to": "t",
      "capacity": 20
    },
    {
      "from": "v4",
      "to": "t",
      "capacity": 4
    }
  ],
  "source": "s",
  "sink": "t",
  "claim": {
    "source_side": [
      "s",
      "v1",
      "v2"
    ],
    "capacity": 26,
    "crossing_edges": [
      3,
      5
    ]
  }
}
```

Response:

```json
{
  "result": {
    "valid": true,
    "reason": null,
    "first_discrepancy": null,
    "cut_capacity": "26",
    "reference_version": "CLRS 4th ed. (2022) ch. 24 Maximum Flow: flow networks, cuts, residual networks, max-flow min-cut theorem; r55-flow-certificate-verifiers@1.0.0"
  }
}
```

## How to call it

### MCP

Connect `https://computefirst.net/mcp` ([setup](/docs#connect)), then call `execute` with:

```json
{
  "id": "network-cut-capacity-check",
  "version": "1.0.0",
  "input": {
    "nodes": [
      "s",
      "v1",
      "v2",
      "v3",
      "v4",
      "t"
    ],
    "edges": [
      {
        "from": "s",
        "to": "v1",
        "capacity": 16
      },
      {
        "from": "s",
        "to": "v2",
        "capacity": 13
      },
      {
        "from": "v2",
        "to": "v1",
        "capacity": 4
      },
      {
        "from": "v1",
        "to": "v3",
        "capacity": 12
      },
      {
        "from": "v3",
        "to": "v2",
        "capacity": 9
      },
      {
        "from": "v2",
        "to": "v4",
        "capacity": 14
      },
      {
        "from": "v4",
        "to": "v3",
        "capacity": 7
      },
      {
        "from": "v3",
        "to": "t",
        "capacity": 20
      },
      {
        "from": "v4",
        "to": "t",
        "capacity": 4
      }
    ],
    "source": "s",
    "sink": "t",
    "claim": {
      "source_side": [
        "s",
        "v1",
        "v2"
      ],
      "capacity": 26,
      "crossing_edges": [
        3,
        5
      ]
    }
  }
}
```

### HTTP (no key)

```sh
curl -X POST https://computefirst.net/v1/tools/network-cut-capacity-check/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"nodes":["s","v1","v2","v3","v4","t"],"edges":[{"from":"s","to":"v1","capacity":16},{"from":"s","to":"v2","capacity":13},{"from":"v2","to":"v1","capacity":4},{"from":"v1","to":"v3","capacity":12},{"from":"v3","to":"v2","capacity":9},{"from":"v2","to":"v4","capacity":14},{"from":"v4","to":"v3","capacity":7},{"from":"v3","to":"t","capacity":20},{"from":"v4","to":"t","capacity":4}],"source":"s","sink":"t","claim":{"source_side":["s","v1","v2"],"capacity":26,"crossing_edges":[3,5]}}'
```

The machine-readable contract is at [/v1/tools/network-cut-capacity-check/versions/1.0.0](/v1/tools/network-cut-capacity-check/versions/1.0.0).

### CLI

```sh
node cli.mjs run network-cut-capacity-check 1.0.0 --input input.json --base-url https://computefirst.net
```

Get the client at [/clients/cli/](/clients/cli/).

## Related tools

- [Network flow feasibility check](/tools/network-flow-feasibility-check): Check a claimed s-t flow for capacity bounds and conservation with exact integers; returns the first bad edge or node.
- [Maximum flow certificate check](/tools/maximum-flow-certificate-check): Check a claimed max flow with a min-cut certificate or residual search; reports the first failure or augmenting path.
- [Graph coloring check](/tools/graph-coloring-check): Check that a node colouring of an undirected graph is proper (and within k colours); returns the first conflict.
- [Graph path weight check](/tools/graph-path-weight-check): Check that a path of edge IDs walks source to target in a weighted digraph and sums to the claimed exact weight.
- [Lp primal feasibility check](/tools/lp-primal-feasibility-check): Check that a claimed LP point satisfies x >= 0 and Ax <= b exactly; report the first violated bound or row.
- [Spanning tree structure check](/tools/spanning-tree-structure-check): Check that claimed edge IDs form a spanning tree of a graph; reports the first missing, repeated or cyclic edge