From 105919383f81fa8f740bec3806e3bf5f2563d048 Mon Sep 17 00:00:00 2001 From: GiggleLiu Date: Wed, 18 Mar 2026 01:22:16 +0800 Subject: [PATCH 1/3] Add plan for #185: MinimumMultiwayCut to ILP --- .../2026-03-18-minimummultiwaycut-to-ilp.md | 99 +++++++++++++++++++ 1 file changed, 99 insertions(+) create mode 100644 docs/plans/2026-03-18-minimummultiwaycut-to-ilp.md diff --git a/docs/plans/2026-03-18-minimummultiwaycut-to-ilp.md b/docs/plans/2026-03-18-minimummultiwaycut-to-ilp.md new file mode 100644 index 000000000..b76ec8045 --- /dev/null +++ b/docs/plans/2026-03-18-minimummultiwaycut-to-ilp.md @@ -0,0 +1,99 @@ +# Plan: MinimumMultiwayCut to ILP Reduction (#185) + +## Overview + +Add a reduction from `MinimumMultiwayCut` to `ILP` using the standard vertex-assignment + edge-cut indicator ILP formulation from Chopra & Owen (1996). + +## Reduction Algorithm + +**Variables** (kn + m binary): +- `y_{iv}` at index `i*n + v` for i in 0..k, v in 0..n: vertex v assigned to component of terminal t_i +- `x_e` at index `kn + e_idx` for each edge: 1 if edge is in the cut + +**Bounds (terminal fixing):** +- `y_{i, t_i} = 1` (terminal t_i is in its own component) +- `y_{j, t_i} = 0` for j != i + +**Constraints:** +1. Partition (n equality): `sum_i y_{iv} = 1` for each vertex v +2. Edge-cut linking (2km inequality): `x_e >= y_{iu} - y_{iv}` and `x_e >= y_{iv} - y_{iu}` for each edge (u,v), each terminal i + +**Objective:** minimize `sum_e w_e * x_e` + +**Solution extraction:** For each edge e, source config[e] = target_solution[kn + e_idx] (the x_e value). + +## Steps + +### Batch 1: Implementation + Tests + Example-DB + +#### Step 1: Implement reduction (`src/rules/minimummultiwaycut_ilp.rs`) + +Follow `binpacking_ilp.rs` pattern: + +```rust +// ReductionMMCToILP struct with fields: target: ILP, n: usize, m: usize, k: usize +// ReductionResult impl: extract_solution reads x_e values at indices kn..kn+m +// #[reduction(overhead = { +// num_vars = "num_terminals * num_vertices + num_edges", +// num_constraints = "num_vertices + 2 * num_terminals * num_edges", +// })] +// impl ReduceTo> for MinimumMultiwayCut +``` + +Key implementation details: +- Use `LinearConstraint::eq` for partition constraints +- Use `LinearConstraint::ge` for edge-cut linking (rearranged: `x_e - y_{iu} + y_{iv} >= 0`) +- Terminal fixing: set bounds via additional equality constraints `y_{i,t_i} = 1` and `y_{j,t_i} = 0` + - Alternative: use `LinearConstraint::eq` with single term to fix variables (simpler than custom bounds) +- Objective: sparse vec of `(kn + e_idx, w_e as f64)` for each edge + +#### Step 2: Register in `src/rules/mod.rs` + +Add under the `#[cfg(feature = "ilp-solver")]` block: +```rust +#[cfg(feature = "ilp-solver")] +pub(crate) mod minimummultiwaycut_ilp; +``` + +#### Step 3: Write unit tests (`src/unit_tests/rules/minimummultiwaycut_ilp.rs`) + +Tests (following `binpacking_ilp.rs` pattern): +1. `test_reduction_creates_valid_ilp` — check num_vars = kn+m, num_constraints = n+2km + terminal fixes, sense = Minimize +2. `test_minimummultiwaycut_to_ilp_closed_loop` — use the canonical 5-vertex example, compare brute-force vs ILP solver +3. `test_triangle_with_3_terminals` — triangle graph, all vertices are terminals, optimal = cut 2 cheapest edges +4. `test_solution_extraction` — manually construct ILP solution, verify extracted config +5. `test_solve_reduced` — use `ILPSolver.solve_reduced(&problem)` +6. `test_two_terminals` — degenerate case with k=2 (min s-t cut) + +#### Step 4: Add canonical example (`src/example_db/rule_builders.rs`) + +Add `canonical_rule_example_specs()` in the reduction file, using the issue's 5-vertex example: +```rust +MinimumMultiwayCut::new( + SimpleGraph::new(5, vec![(0,1),(1,2),(2,3),(3,4),(0,4),(1,3)]), + vec![0, 2, 4], + vec![2, 3, 1, 2, 4, 5], +) +``` +Register in `src/rules/mod.rs` `canonical_rule_example_specs()` function under the `#[cfg(feature = "ilp-solver")]` block. + +### Batch 2: Paper Documentation + +#### Step 5: Document in `docs/paper/reductions.typ` + +Add `reduction-rule("MinimumMultiwayCut", "ILP", example: true, ...)` entry. Include: +- Theorem statement: the ILP formulation with variable/constraint counts +- Proof: construction details (variable mapping, constraints, terminal fixing, objective), correctness argument, solution extraction +- Worked example from the canonical fixture JSON + +### Batch 3: Export + Verify + +#### Step 6: Regenerate exports and verify + +```bash +cargo run --example export_graph +cargo run --example export_schemas +make test +make clippy +make paper +``` From c2e6b2ea964ea60702f4250406d15c8b47b2e965 Mon Sep 17 00:00:00 2001 From: GiggleLiu Date: Wed, 18 Mar 2026 01:41:42 +0800 Subject: [PATCH 2/3] Implement MinimumMultiwayCut to ILP reduction (#185) Add vertex-assignment + edge-cut indicator ILP formulation (Chopra & Owen 1996) with kn+m binary variables and n+2km+k^2 constraints. Includes 6 unit tests (closed-loop, triangle, 2-terminal, extraction, solve_reduced), canonical example-db entry, paper documentation, and bib reference. --- docs/paper/reductions.typ | 16 ++ docs/paper/references.bib | 10 ++ .../2026-03-18-minimummultiwaycut-to-ilp.md | 99 ------------ docs/src/reductions/reduction_graph.json | 15 ++ problemreductions-cli/src/commands/create.rs | 1 - src/example_db/fixtures/examples.json | 27 ++-- src/rules/minimummultiwaycut_ilp.rs | 144 ++++++++++++++++++ src/rules/mod.rs | 3 + .../rules/minimummultiwaycut_ilp.rs | 139 +++++++++++++++++ 9 files changed, 341 insertions(+), 113 deletions(-) delete mode 100644 docs/plans/2026-03-18-minimummultiwaycut-to-ilp.md create mode 100644 src/rules/minimummultiwaycut_ilp.rs create mode 100644 src/unit_tests/rules/minimummultiwaycut_ilp.rs diff --git a/docs/paper/reductions.typ b/docs/paper/reductions.typ index 1c0e5f785..5fd657a19 100644 --- a/docs/paper/reductions.typ +++ b/docs/paper/reductions.typ @@ -3329,6 +3329,22 @@ The following reductions to Integer Linear Programming are straightforward formu _Solution extraction._ Collect pairs $(j_1, j_2)$ with $m_(j_1, j_2) = 1$, sort by $j_1$, and read the characters. ] +#reduction-rule("MinimumMultiwayCut", "ILP")[ + The vertex-assignment + edge-cut indicator formulation @chopra1996 introduces binary variables for vertex-to-component membership and edge-cut indicators. Terminal vertices are fixed to their own components, partition constraints ensure every vertex belongs to exactly one component, and linking inequalities force the cut indicator on whenever an edge's endpoints are in different components. +][ + _Construction._ Given graph $G = (V, E, w)$ with $n = |V|$ vertices, $m = |E|$ edges, edge weights $w_e > 0$, and $k$ terminals $T = {t_0, dots, t_(k-1)}$: + + _Variables:_ (1) $y_(i v) in {0, 1}$ for $i in {0, dots, k-1}$, $v in V$: vertex $v$ belongs to the component of terminal $t_i$. (2) $x_e in {0, 1}$ for $e in E$: edge $e$ is in the cut. Total: $k n + m$ variables. + + _Constraints:_ (1) Terminal fixing: $y_(i, t_i) = 1$ for each $i$ (terminal $t_i$ is in its own component); $y_(j, t_i) = 0$ for $j eq.not i$ (each terminal excluded from other components). (2) Partition: $sum_(i=0)^(k-1) y_(i v) = 1$ for each $v in V$ (each vertex in exactly one component). (3) Edge-cut linking: for each edge $e = (u, v)$ and each terminal $i$: $x_e gt.eq y_(i u) - y_(i v)$ and $x_e gt.eq y_(i v) - y_(i u)$ (force $x_e = 1$ when endpoints are in different components). Total: $k^2 + n + 2 k m$ constraints. + + _Objective:_ Minimize $sum_(e in E) w_e dot x_e$. + + _Correctness._ ($arrow.r.double$) A multiway cut $C$ partitions $V$ into $k$ components, one per terminal. Setting $y_(i v) = 1$ iff $v$ is in $t_i$'s component and $x_e = 1$ iff $e in C$ satisfies all constraints: partition by construction, terminal fixing by definition, and linking because any edge with endpoints in different components is in $C$. The objective equals the cut weight. ($arrow.l.double$) Any feasible ILP solution defines a valid partition (by constraint (2)) separating all terminals (by constraint (1)). The linking constraints (3) force $x_e = 1$ for all cross-component edges, so the objective is at least the multiway cut weight; minimization ensures optimality. + + _Solution extraction._ For each edge $e$ at index $"idx"$, read $x_e = x^*_(k n + "idx"))$. The source configuration is $"config"[e] = x_e$ (1 = cut, 0 = keep). +] + == Unit Disk Mapping #reduction-rule("MaximumIndependentSet", "KingsSubgraph")[ diff --git a/docs/paper/references.bib b/docs/paper/references.bib index 34fc92a17..c643d8315 100644 --- a/docs/paper/references.bib +++ b/docs/paper/references.bib @@ -734,3 +734,13 @@ @article{papadimitriou1982 year = {1982}, doi = {10.1145/322307.322309} } + +@article{chopra1996, + author = {Sunil Chopra and Jonathan H. Owen}, + title = {Extended formulations for the A-cut problem}, + journal = {Mathematical Programming}, + volume = {73}, + pages = {7--30}, + year = {1996}, + doi = {10.1007/BF02592096} +} diff --git a/docs/plans/2026-03-18-minimummultiwaycut-to-ilp.md b/docs/plans/2026-03-18-minimummultiwaycut-to-ilp.md deleted file mode 100644 index b76ec8045..000000000 --- a/docs/plans/2026-03-18-minimummultiwaycut-to-ilp.md +++ /dev/null @@ -1,99 +0,0 @@ -# Plan: MinimumMultiwayCut to ILP Reduction (#185) - -## Overview - -Add a reduction from `MinimumMultiwayCut` to `ILP` using the standard vertex-assignment + edge-cut indicator ILP formulation from Chopra & Owen (1996). - -## Reduction Algorithm - -**Variables** (kn + m binary): -- `y_{iv}` at index `i*n + v` for i in 0..k, v in 0..n: vertex v assigned to component of terminal t_i -- `x_e` at index `kn + e_idx` for each edge: 1 if edge is in the cut - -**Bounds (terminal fixing):** -- `y_{i, t_i} = 1` (terminal t_i is in its own component) -- `y_{j, t_i} = 0` for j != i - -**Constraints:** -1. Partition (n equality): `sum_i y_{iv} = 1` for each vertex v -2. Edge-cut linking (2km inequality): `x_e >= y_{iu} - y_{iv}` and `x_e >= y_{iv} - y_{iu}` for each edge (u,v), each terminal i - -**Objective:** minimize `sum_e w_e * x_e` - -**Solution extraction:** For each edge e, source config[e] = target_solution[kn + e_idx] (the x_e value). - -## Steps - -### Batch 1: Implementation + Tests + Example-DB - -#### Step 1: Implement reduction (`src/rules/minimummultiwaycut_ilp.rs`) - -Follow `binpacking_ilp.rs` pattern: - -```rust -// ReductionMMCToILP struct with fields: target: ILP, n: usize, m: usize, k: usize -// ReductionResult impl: extract_solution reads x_e values at indices kn..kn+m -// #[reduction(overhead = { -// num_vars = "num_terminals * num_vertices + num_edges", -// num_constraints = "num_vertices + 2 * num_terminals * num_edges", -// })] -// impl ReduceTo> for MinimumMultiwayCut -``` - -Key implementation details: -- Use `LinearConstraint::eq` for partition constraints -- Use `LinearConstraint::ge` for edge-cut linking (rearranged: `x_e - y_{iu} + y_{iv} >= 0`) -- Terminal fixing: set bounds via additional equality constraints `y_{i,t_i} = 1` and `y_{j,t_i} = 0` - - Alternative: use `LinearConstraint::eq` with single term to fix variables (simpler than custom bounds) -- Objective: sparse vec of `(kn + e_idx, w_e as f64)` for each edge - -#### Step 2: Register in `src/rules/mod.rs` - -Add under the `#[cfg(feature = "ilp-solver")]` block: -```rust -#[cfg(feature = "ilp-solver")] -pub(crate) mod minimummultiwaycut_ilp; -``` - -#### Step 3: Write unit tests (`src/unit_tests/rules/minimummultiwaycut_ilp.rs`) - -Tests (following `binpacking_ilp.rs` pattern): -1. `test_reduction_creates_valid_ilp` — check num_vars = kn+m, num_constraints = n+2km + terminal fixes, sense = Minimize -2. `test_minimummultiwaycut_to_ilp_closed_loop` — use the canonical 5-vertex example, compare brute-force vs ILP solver -3. `test_triangle_with_3_terminals` — triangle graph, all vertices are terminals, optimal = cut 2 cheapest edges -4. `test_solution_extraction` — manually construct ILP solution, verify extracted config -5. `test_solve_reduced` — use `ILPSolver.solve_reduced(&problem)` -6. `test_two_terminals` — degenerate case with k=2 (min s-t cut) - -#### Step 4: Add canonical example (`src/example_db/rule_builders.rs`) - -Add `canonical_rule_example_specs()` in the reduction file, using the issue's 5-vertex example: -```rust -MinimumMultiwayCut::new( - SimpleGraph::new(5, vec![(0,1),(1,2),(2,3),(3,4),(0,4),(1,3)]), - vec![0, 2, 4], - vec![2, 3, 1, 2, 4, 5], -) -``` -Register in `src/rules/mod.rs` `canonical_rule_example_specs()` function under the `#[cfg(feature = "ilp-solver")]` block. - -### Batch 2: Paper Documentation - -#### Step 5: Document in `docs/paper/reductions.typ` - -Add `reduction-rule("MinimumMultiwayCut", "ILP", example: true, ...)` entry. Include: -- Theorem statement: the ILP formulation with variable/constraint counts -- Proof: construction details (variable mapping, constraints, terminal fixing, objective), correctness argument, solution extraction -- Worked example from the canonical fixture JSON - -### Batch 3: Export + Verify - -#### Step 6: Regenerate exports and verify - -```bash -cargo run --example export_graph -cargo run --example export_schemas -make test -make clippy -make paper -``` diff --git a/docs/src/reductions/reduction_graph.json b/docs/src/reductions/reduction_graph.json index 756c52b70..ac99fd408 100644 --- a/docs/src/reductions/reduction_graph.json +++ b/docs/src/reductions/reduction_graph.json @@ -1265,6 +1265,21 @@ ], "doc_path": "rules/minimumdominatingset_ilp/index.html" }, + { + "source": 47, + "target": 16, + "overhead": [ + { + "field": "num_vars", + "formula": "num_terminals * num_vertices + num_edges" + }, + { + "field": "num_constraints", + "formula": "num_vertices + 2 * num_terminals * num_edges + num_terminals * num_terminals" + } + ], + "doc_path": "rules/minimummultiwaycut_ilp/index.html" + }, { "source": 48, "target": 16, diff --git a/problemreductions-cli/src/commands/create.rs b/problemreductions-cli/src/commands/create.rs index 973005907..f58d597f7 100644 --- a/problemreductions-cli/src/commands/create.rs +++ b/problemreductions-cli/src/commands/create.rs @@ -1639,7 +1639,6 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> { emit_problem_output(&output, out) } - /// Reject non-unit weights when the resolved variant uses `weight=One`. fn reject_nonunit_weights_for_one_variant( canonical: &str, diff --git a/src/example_db/fixtures/examples.json b/src/example_db/fixtures/examples.json index d8dd0fb6e..070c5448f 100644 --- a/src/example_db/fixtures/examples.json +++ b/src/example_db/fixtures/examples.json @@ -54,18 +54,18 @@ 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{"source":{"problem":"TravelingSalesman","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"edge_weights":[10,15,20,35,25,30],"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,2,null],[0,3,null],[1,2,null],[1,3,null],[2,3,null]],"node_holes":[],"nodes":[null,null,null,null]}}}},"target":{"problem":"ILP","variant":{"variable":"bool"},"instance":{"constraints":[{"cmp":"Eq","rhs":1.0,"terms":[[0,1.0],[1,1.0],[2,1.0],[3,1.0]]},{"cmp":"Eq","rhs":1.0,"terms":[[4,1.0],[5,1.0],[6,1.0],[7,1.0]]},{"cmp":"Eq","rhs":1.0,"terms":[[8,1.0],[9,1.0],[10,1.0],[11,1.0]]},{"cmp":"Eq","rhs":1.0,"terms":[[12,1.0],[13,1.0],[14,1.0],[15,1.0]]},{"cmp":"Eq","rhs":1.0,"terms":[[0,1.0],[4,1.0],[8,1.0],[12,1.0]]},{"cmp":"Eq","rhs":1.0,"terms":[[1,1.0],[5,1.0],[9,1.0],[13,1.0]]},{"cmp":"Eq","rhs":1.0,"terms":[[2,1.0],[6,1.0],[10,1.0],[14,1.0]]},{"cmp":"Eq","rhs":1.0,"terms":[[3,1.0],[7,1.0],[11,1.0],[15,1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[16,1.0],[0,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[16,1.0],[5,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[16,1.0],[0,-1.0],[5,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[17,1.0],[4,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[17,1.0],[1,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[17,1.0],[4,-1.0],[1,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[18,1.0],[1,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[18,1.0],[6,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[18,1.0],[1,-1.0],[6,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[19,1.0],[5,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[19,1.0],[2,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[19,1.0],[5,-1.0],[2,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[20,1.0],[2,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[20,1.0],[7,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[20,1.0],[2,-1.0],[7,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[21,1.0],[6,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[21,1.0],[3,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[21,1.0],[6,-1.0],[3,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[22,1.0],[3,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[22,1.0],[4,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[22,1.0],[3,-1.0],[4,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[23,1.0],[7,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[23,1.0],[0,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[23,1.0],[7,-1.0],[0,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[24,1.0],[0,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[24,1.0],[9,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[24,1.0],[0,-1.0],[9,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[25,1.0],[8,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[25,1.0],[1,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[25,1.0],[8,-1.0],[1,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[26,1.0],[1,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[26,1.0],[10,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[26,1.0],[1,-1.0],[10,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[27,1.0],[9,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[27,1.0],[2,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[27,1.0],[9,-1.0],[2,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[28,1.0],[2,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[28,1.0],[11,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[28,1.0],[2,-1.0],[11,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[29,1.0],[10,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[29,1.0],[3,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[29,1.0],[10,-1.0],[3,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[30,1.0],[3,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[30,1.0],[8,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[30,1.0],[3,-1.0],[8,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[31,1.0],[11,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[31,1.0],[0,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[31,1.0],[11,-1.0],[0,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[32,1.0],[0,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[32,1.0],[13,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[32,1.0],[0,-1.0],[13,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[33,1.0],[12,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[33,1.0],[1,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[33,1.0],[12,-1.0],[1,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[34,1.0],[1,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[34,1.0],[14,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[34,1.0],[1,-1.0],[14,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[35,1.0],[13,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[35,1.0],[2,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[35,1.0],[13,-1.0],[2,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[36,1.0],[2,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[36,1.0],[15,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[36,1.0],[2,-1.0],[15,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[37,1.0],[14,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[37,1.0],[3,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[37,1.0],[14,-1.0],[3,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[38,1.0],[3,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[38,1.0],[12,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[38,1.0],[3,-1.0],[12,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[39,1.0],[15,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[39,1.0],[0,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[39,1.0],[15,-1.0],[0,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[40,1.0],[4,-1.0]]},{"cmp":"Le","rhs":0.0,"terms":[[40,1.0],[9,-1.0]]},{"cmp":"Ge","rhs":-1.0,"terms":[[40,1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+ {"source":{"problem":"TravelingSalesman","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"edge_weights":[1,2,3],"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,2,null],[1,2,null]],"node_holes":[],"nodes":[null,null,null]}}}},"target":{"problem":"QUBO","variant":{"weight":"f64"},"instance":{"matrix":[[-14.0,14.0,14.0,14.0,1.0,1.0,14.0,2.0,2.0],[0.0,-14.0,14.0,1.0,14.0,1.0,2.0,14.0,2.0],[0.0,0.0,-14.0,1.0,1.0,14.0,2.0,2.0,14.0],[0.0,0.0,0.0,-14.0,14.0,14.0,14.0,3.0,3.0],[0.0,0.0,0.0,0.0,-14.0,14.0,3.0,14.0,3.0],[0.0,0.0,0.0,0.0,0.0,-14.0,3.0,3.0,14.0],[0.0,0.0,0.0,0.0,0.0,0.0,-14.0,14.0,14.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,-14.0,14.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,-14.0]],"num_vars":9}},"solutions":[{"source_config":[1,1,1],"target_config":[0,0,1,1,0,0,0,1,0]}]} ] } diff --git a/src/rules/minimummultiwaycut_ilp.rs b/src/rules/minimummultiwaycut_ilp.rs new file mode 100644 index 000000000..211ce2344 --- /dev/null +++ b/src/rules/minimummultiwaycut_ilp.rs @@ -0,0 +1,144 @@ +//! Reduction from MinimumMultiwayCut to ILP (Integer Linear Programming). +//! +//! Uses the standard vertex-assignment + edge-cut indicator formulation +//! (Chopra & Owen, 1996): +//! - Variables: `y_{iv}` (vertex v in component i) + `x_e` (edge e in cut), all binary +//! - Constraints: partition (each vertex in exactly one component) + edge-cut linking +//! - Objective: minimize total weight of cut edges + +use crate::models::algebraic::{LinearConstraint, ObjectiveSense, ILP}; +use crate::models::graph::MinimumMultiwayCut; +use crate::reduction; +use crate::rules::traits::{ReduceTo, ReductionResult}; +use crate::topology::{Graph, SimpleGraph}; + +/// Result of reducing MinimumMultiwayCut to ILP. +/// +/// Variable layout (all binary): +/// - `y_{iv}` for i=0..k-1, v=0..n-1: vertex v assigned to component of terminal t_i +/// (index: i*n + v) +/// - `x_e` for e=0..m-1: edge e is in the cut (index: k*n + e) +/// +/// Total: kn + m variables. +#[derive(Debug, Clone)] +pub struct ReductionMMCToILP { + target: ILP, + /// Number of vertices in the source graph. + n: usize, + /// Number of edges in the source graph. + m: usize, + /// Number of terminals. + k: usize, +} + +impl ReductionResult for ReductionMMCToILP { + type Source = MinimumMultiwayCut; + type Target = ILP; + + fn target_problem(&self) -> &ILP { + &self.target + } + + /// Extract solution from ILP back to MinimumMultiwayCut. + /// + /// For each edge e, source config[e] = target_solution[k*n + e] (the x_e variable). + fn extract_solution(&self, target_solution: &[usize]) -> Vec { + let offset = self.k * self.n; + (0..self.m).map(|e| target_solution[offset + e]).collect() + } +} + +#[reduction( + overhead = { + num_vars = "num_terminals * num_vertices + num_edges", + num_constraints = "num_vertices + 2 * num_terminals * num_edges + num_terminals * num_terminals", + } +)] +impl ReduceTo> for MinimumMultiwayCut { + type Result = ReductionMMCToILP; + + fn reduce_to(&self) -> Self::Result { + let n = self.num_vertices(); + let m = self.num_edges(); + let k = self.num_terminals(); + let terminals = self.terminals(); + let edges = self.graph().edges(); + let weights = self.edge_weights(); + let num_vars = k * n + m; + + // Terminal fixing constraints: k constraints for y_{i,t_i} = 1, + // and k*(k-1) constraints for y_{j,t_i} = 0 where j != i. + // Total terminal fixes: k + k*(k-1) = k^2. + let num_terminal_fixes = k * k; + let num_constraints = n + 2 * k * m + num_terminal_fixes; + let mut constraints = Vec::with_capacity(num_constraints); + + // Terminal fixing: y_{i, t_i} = 1 for each terminal i + for (i, &t) in terminals.iter().enumerate() { + constraints.push(LinearConstraint::eq(vec![(i * n + t, 1.0)], 1.0)); + } + + // Terminal fixing: y_{j, t_i} = 0 for j != i + for (i, &t) in terminals.iter().enumerate() { + for j in 0..k { + if j != i { + constraints.push(LinearConstraint::eq(vec![(j * n + t, 1.0)], 0.0)); + } + } + } + + // Partition constraints: sum_i y_{iv} = 1 for each vertex v + for v in 0..n { + let terms: Vec<(usize, f64)> = (0..k).map(|i| (i * n + v, 1.0)).collect(); + constraints.push(LinearConstraint::eq(terms, 1.0)); + } + + // Edge-cut linking constraints: for each edge e=(u,v) and each terminal i: + // x_e >= y_{iu} - y_{iv} => x_e - y_{iu} + y_{iv} >= 0 + // x_e >= y_{iv} - y_{iu} => x_e + y_{iu} - y_{iv} >= 0 + for (e_idx, (u, v)) in edges.iter().enumerate() { + let x_var = k * n + e_idx; + for i in 0..k { + let y_iu = i * n + u; + let y_iv = i * n + v; + // x_e - y_{iu} + y_{iv} >= 0 + constraints.push(LinearConstraint::ge( + vec![(x_var, 1.0), (y_iu, -1.0), (y_iv, 1.0)], + 0.0, + )); + // x_e + y_{iu} - y_{iv} >= 0 + constraints.push(LinearConstraint::ge( + vec![(x_var, 1.0), (y_iu, 1.0), (y_iv, -1.0)], + 0.0, + )); + } + } + + // Objective: minimize sum_e w_e * x_e + let objective: Vec<(usize, f64)> = weights + .iter() + .enumerate() + .map(|(e_idx, w)| (k * n + e_idx, *w as f64)) + .collect(); + + let target = ILP::new(num_vars, constraints, objective, ObjectiveSense::Minimize); + + ReductionMMCToILP { target, n, m, k } + } +} + +#[cfg(feature = "example-db")] +pub(crate) fn canonical_rule_example_specs() -> Vec { + vec![crate::example_db::specs::RuleExampleSpec { + id: "minimummultiwaycut_to_ilp", + build: || { + let graph = SimpleGraph::new(5, vec![(0, 1), (1, 2), (2, 3), (3, 4), (0, 4), (1, 3)]); + let problem = MinimumMultiwayCut::new(graph, vec![0, 2, 4], vec![2, 3, 1, 2, 4, 5]); + crate::example_db::specs::direct_ilp_example::<_, bool, _>(problem, |_, _| true) + }, + }] +} + +#[cfg(test)] +#[path = "../unit_tests/rules/minimummultiwaycut_ilp.rs"] +mod tests; diff --git a/src/rules/mod.rs b/src/rules/mod.rs index 6fc0ae7a0..a502232dd 100644 --- a/src/rules/mod.rs +++ b/src/rules/mod.rs @@ -64,6 +64,8 @@ pub(crate) mod maximumsetpacking_ilp; #[cfg(feature = "ilp-solver")] pub(crate) mod minimumdominatingset_ilp; #[cfg(feature = "ilp-solver")] +pub(crate) mod minimummultiwaycut_ilp; +#[cfg(feature = "ilp-solver")] pub(crate) mod minimumsetcovering_ilp; #[cfg(feature = "ilp-solver")] pub(crate) mod qubo_ilp; @@ -110,6 +112,7 @@ pub(crate) fn canonical_rule_example_specs() -> Vec MinimumMultiwayCut { + let graph = SimpleGraph::new(5, vec![(0, 1), (1, 2), (2, 3), (3, 4), (0, 4), (1, 3)]); + MinimumMultiwayCut::new(graph, vec![0, 2, 4], vec![2, 3, 1, 2, 4, 5]) +} + +#[test] +fn test_reduction_creates_valid_ilp() { + let problem = canonical_instance(); + let reduction: ReductionMMCToILP = ReduceTo::>::reduce_to(&problem); + let ilp = reduction.target_problem(); + + let k = 3; + let n = 5; + let m = 6; + // kn + m = 21 variables + assert_eq!(ilp.num_vars, k * n + m); + // n + 2km + k^2 = 5 + 36 + 9 = 50 constraints + assert_eq!(ilp.constraints.len(), n + 2 * k * m + k * k); + assert_eq!(ilp.sense, ObjectiveSense::Minimize); +} + +#[test] +fn test_minimummultiwaycut_to_ilp_closed_loop() { + let problem = canonical_instance(); + let reduction: ReductionMMCToILP = ReduceTo::>::reduce_to(&problem); + let ilp = reduction.target_problem(); + + let bf = BruteForce::new(); + let ilp_solver = ILPSolver::new(); + + // Solve original with brute force + let bf_solutions = bf.find_all_best(&problem); + let bf_obj = problem.evaluate(&bf_solutions[0]); + + // Solve via ILP + let ilp_solution = ilp_solver.solve(ilp).expect("ILP should be solvable"); + let extracted = reduction.extract_solution(&ilp_solution); + let ilp_obj = problem.evaluate(&extracted); + + // Optimal cut cost is 8 + assert_eq!(bf_obj, SolutionSize::Valid(8)); + assert_eq!(ilp_obj, SolutionSize::Valid(8)); +} + +#[test] +fn test_triangle_with_3_terminals() { + // Triangle: 3 vertices, all terminals, edges: (0,1)=1, (1,2)=2, (0,2)=3 + // All 3 edges must be cut to separate every terminal pair (complete graph). + // Optimal cost = 1 + 2 + 3 = 6 + let graph = SimpleGraph::new(3, vec![(0, 1), (1, 2), (0, 2)]); + let problem = MinimumMultiwayCut::new(graph, vec![0, 1, 2], vec![1, 2, 3]); + + let reduction: ReductionMMCToILP = ReduceTo::>::reduce_to(&problem); + let ilp = reduction.target_problem(); + + let ilp_solver = ILPSolver::new(); + let ilp_solution = ilp_solver.solve(ilp).expect("ILP should be solvable"); + let extracted = reduction.extract_solution(&ilp_solution); + + let obj = problem.evaluate(&extracted); + assert_eq!(obj, SolutionSize::Valid(6)); +} + +#[test] +fn test_two_terminals() { + // Path: 0--1--2, terminals {0, 2}, weights [1, 2] + // Optimal min s-t cut: cut edge (0,1) with cost 1 + let graph = SimpleGraph::new(3, vec![(0, 1), (1, 2)]); + let problem = MinimumMultiwayCut::new(graph, vec![0, 2], vec![1, 2]); + + let reduction: ReductionMMCToILP = ReduceTo::>::reduce_to(&problem); + let ilp = reduction.target_problem(); + + let ilp_solver = ILPSolver::new(); + let ilp_solution = ilp_solver.solve(ilp).expect("ILP should be solvable"); + let extracted = reduction.extract_solution(&ilp_solution); + + let obj = problem.evaluate(&extracted); + assert_eq!(obj, SolutionSize::Valid(1)); +} + +#[test] +fn test_solution_extraction() { + let problem = canonical_instance(); + let reduction: ReductionMMCToILP = ReduceTo::>::reduce_to(&problem); + + let k = 3; + let n = 5; + let m = 6; + let num_vars = k * n + m; + + // Manually construct an ILP solution representing the optimal partition: + // V_0 = {0}, V_1 = {1, 2, 3}, V_2 = {4} + // Cut edges: (0,1)=idx 0, (3,4)=idx 3, (0,4)=idx 4 + let mut ilp_solution = vec![0usize; num_vars]; + + // y_{0,v}: component 0 assignments (indices 0..5) + ilp_solution[0] = 1; // y_{0,0} = 1 (vertex 0 in component 0) + + // y_{1,v}: component 1 assignments (indices 5..10) + ilp_solution[5 + 1] = 1; // y_{1,1} = 1 + ilp_solution[5 + 2] = 1; // y_{1,2} = 1 + ilp_solution[5 + 3] = 1; // y_{1,3} = 1 + + // y_{2,v}: component 2 assignments (indices 10..15) + ilp_solution[10 + 4] = 1; // y_{2,4} = 1 + + // x_e: cut indicators (indices 15..21) + ilp_solution[15] = 1; // edge (0,1) cut + ilp_solution[15 + 3] = 1; // edge (3,4) cut + ilp_solution[15 + 4] = 1; // edge (0,4) cut + + let extracted = reduction.extract_solution(&ilp_solution); + assert_eq!(extracted, vec![1, 0, 0, 1, 1, 0]); + + let obj = problem.evaluate(&extracted); + assert_eq!(obj, SolutionSize::Valid(8)); +} + +#[test] +fn test_solve_reduced() { + let problem = canonical_instance(); + + let ilp_solver = ILPSolver::new(); + let solution = ilp_solver + .solve_reduced(&problem) + .expect("solve_reduced should work"); + + assert!(problem.evaluate(&solution).is_valid()); + assert_eq!(problem.evaluate(&solution), SolutionSize::Valid(8)); +} From 019be1749fa8fb1afcb6e23e53e0687b5ac8ba9c Mon Sep 17 00:00:00 2001 From: zazabap Date: Wed, 18 Mar 2026 12:31:09 +0000 Subject: [PATCH 3/3] fix: remove extra closing parenthesis in paper reduction-rule entry Co-Authored-By: Claude Opus 4.6 (1M context) --- docs/paper/reductions.typ | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/paper/reductions.typ b/docs/paper/reductions.typ index f0ffb280b..4f3cb6ad2 100644 --- a/docs/paper/reductions.typ +++ b/docs/paper/reductions.typ @@ -3819,7 +3819,7 @@ The following reductions to Integer Linear Programming are straightforward formu _Correctness._ ($arrow.r.double$) A multiway cut $C$ partitions $V$ into $k$ components, one per terminal. Setting $y_(i v) = 1$ iff $v$ is in $t_i$'s component and $x_e = 1$ iff $e in C$ satisfies all constraints: partition by construction, terminal fixing by definition, and linking because any edge with endpoints in different components is in $C$. The objective equals the cut weight. ($arrow.l.double$) Any feasible ILP solution defines a valid partition (by constraint (2)) separating all terminals (by constraint (1)). The linking constraints (3) force $x_e = 1$ for all cross-component edges, so the objective is at least the multiway cut weight; minimization ensures optimality. - _Solution extraction._ For each edge $e$ at index $"idx"$, read $x_e = x^*_(k n + "idx"))$. The source configuration is $"config"[e] = x_e$ (1 = cut, 0 = keep). + _Solution extraction._ For each edge $e$ at index $"idx"$, read $x_e = x^*_(k n + "idx")$. The source configuration is $"config"[e] = x_e$ (1 = cut, 0 = keep). ] == Unit Disk Mapping