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Add plan for #363: [Rule] PARTITION to INTEGRAL FLOW WITH MULTIPLIERS
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‎docs/plans/2026-05-26-partition-to-integral-flow-with-multipliers.md‎

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# Issue 363 Plan: Partition -> IntegralFlowWithMultipliers
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Issue: [#363](https://github.com/CodingThrust/problem-reductions/issues/363)
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Title: `[Rule] PARTITION to INTEGRAL FLOW WITH MULTIPLIERS`
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## Objective
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Implement the witness-preserving reduction `Partition -> IntegralFlowWithMultipliers` using Sahni's 1974 multiplier-flow gadget with the relay bottleneck fix documented in the issue body and comments. The rule must map:
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- even-total `Partition` instances to a relay network whose sink inflow is forced to equal `S / 2`, and
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- odd-total instances to a fixed infeasible `IntegralFlowWithMultipliers` instance.
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Verification mode: default. No `--no-verify`.
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## Reference Notes
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- Primary source: Sartaj Sahni, *Computationally Related Problems*, SIAM J. Comput. 3(4):262-279, 1974, Section 2.2 / Fig. 2.2.1 (`sum of subsets -> N(i)`).
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- Catalog source: Garey and Johnson, ND33, `Integral Flow With Multipliers`.
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- Issue comments already resolved the earlier false-positive construction by adding relay vertex `w` and bottleneck arc `(w, t)` with capacity `S / 2`.
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## Action Pipeline
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This plan follows `.claude/skills/add-rule/SKILL.md` Steps 1-7.
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## Batch 1: Steps 1-5.5 (verification, implementation, tests, example-db)
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### 1. Mathematical verification
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- Re-state the reduction precisely in repository semantics:
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- Source: `Partition`
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- Target: `IntegralFlowWithMultipliers`
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- Witness on source: binary subset vector over `sizes`
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- Witness on target: integral arc-flow vector in graph arc order
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- Verify the two branches:
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- odd total `S`: fixed 3-vertex NO instance with `h(u) = 2`, capacities `(1, 1)`, `R = 1`
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- even total `S`: vertices `s, v_1, ..., v_n, w, t`; arcs `(s, v_i)`, `(v_i, w)`, `(w, t)`; multipliers `h(v_i) = a_i`, `h(w) = 1`; requirement `R = S / 2`
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- Lock the extraction rule:
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- read the `n` source-item arcs `(s, v_i)`
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- map `flow = 1` to selected item and `flow = 0` to unselected item
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- odd branch returns the all-zero source config
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### 2. Implement the reduction
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- Add `src/rules/partition_integralflowwithmultipliers.rs`.
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- Implement `ReductionPartitionToIntegralFlowWithMultipliers` with:
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- `target: IntegralFlowWithMultipliers`
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- `source_n: usize`
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- `item_arc_count: usize` so extraction can distinguish even vs odd branch reliably
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- Implement `ReduceTo<IntegralFlowWithMultipliers> for Partition`.
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- Construction details:
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- even branch:
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- vertex numbering `0 = s`, `1..=n = v_i`, `n + 1 = w`, `n + 2 = t`
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- arcs in deterministic order: all `(s, v_i)`, then all `(v_i, w)`, then `(w, t)`
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- capacities: `1`, then `a_i`, then `S / 2`
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- multipliers: source/sink placeholders `1`, item multipliers `a_i`, relay multiplier `1`
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- requirement `S / 2`
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- odd branch:
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- fixed graph `s -> u -> t`
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- capacities `[1, 1]`
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- multipliers `[1, 2, 1]`
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- requirement `1`
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- Add exact overhead metadata:
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- `num_vertices = "num_elements + 3"`
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- `num_arcs = "2 * num_elements + 1"`
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- `max_capacity = "total_sum"`
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- `requirement = "total_sum"`
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Notes:
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- these are valid asymptotic upper bounds across both branches; the odd branch is constant size
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- `total_sum` safely upper-bounds both `S / 2` and `max_i a_i`
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### 3. Register in `src/rules/mod.rs`
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- Add `mod partition_integralflowwithmultipliers;`.
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### 4. Write unit tests
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- Add `src/unit_tests/rules/partition_integralflowwithmultipliers.rs`.
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- Required coverage:
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- closed-loop YES instance using brute force on target and round-tripping to source
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- even-sum NO instance `{3, 5}` proving the bottleneck removes the earlier false positive
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- odd-sum NO instance `{1, 2}` proving the fixed NO target is infeasible
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- structure test on the worked example `{2, 3, 4, 5, 6, 4}`:
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- vertex count `9`
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- arc order and capacities
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- multipliers
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- requirement `12`
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- extraction test from a hand-written feasible target flow
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- Reuse `assert_satisfaction_round_trip_from_satisfaction_target` if it fits the witness-preserving pattern.
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### 5. Add canonical example to `example_db`
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- Add a builder in `src/example_db/rule_builders.rs` with id `partition_to_integralflowwithmultipliers`.
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- Use the issue's tutorial instance `A = {2, 3, 4, 5, 6, 4}` and the canonical half-sum witness selecting `{2, 4, 6}`.
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- Ensure the target witness matches the rule's deterministic arc order:
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- source arcs: `[1, 0, 1, 0, 1, 0]`
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- relay arcs: `[2, 0, 4, 0, 6, 0]`
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- bottleneck arc: `[12]`
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### 5.5. Local rule-level verification before paper
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- Run focused tests for:
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- model feasibility expectations
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- new rule unit tests
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- example-db lookup if needed
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- Fix any witness-ordering or feasibility mismatches before touching paper.
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## Batch 2: Step 6 and Step 7 (paper, exports, fixtures, final verification)
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### 6. Document in paper
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- Add a `reduction-rule("Partition", "IntegralFlowWithMultipliers", ...)` entry to `docs/paper/reductions.typ`.
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- Include:
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- construction summary citing Sahni 1974 / Garey-Johnson ND33
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- correctness proof with the exact-equality argument:
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- cap `(w, t) <= S / 2`
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- sink requirement `>= S / 2`
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- therefore sink inflow `= S / 2`
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- solution extraction from the unit-capacity source-item arcs
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- explicit odd-total preprocessing branch
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- Add a worked example derived from the canonical example fixture, starting with `pred-commands()`.
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### 7. Regenerate exports and verify
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- Run the required generators after the rule and paper are in place:
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- `cargo run --example export_graph`
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- `cargo run --example export_schemas`
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- `make regenerate-fixtures`
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- Run default verification commands without `--no-verify`:
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- at minimum `make test clippy`
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- `make paper`
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- Inspect `git status --short` and stage only intended tracked outputs. Ignore generated `docs/src/reductions/` exports if they appear untracked/ignored.
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## Expected Deliverables
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- New reduction source file and tests
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- Rule registration
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- Canonical example-db entry and regenerated fixture data
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- Paper entry for `Partition -> IntegralFlowWithMultipliers`
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- Updated reduction graph / schema exports
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- Clean implementation commit(s), plan-file removal commit, PR comment, and pushed branch

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