feat: add problem_size() to Problem trait with validation (#76)

* feat: add problem_size() to Problem trait (compile-breaking)

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* feat: implement problem_size() for all graph problems

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* feat: implement problem_size() for SAT problems

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* feat: implement problem_size() for set problems

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* feat: implement problem_size() for optimization problems

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* feat: implement problem_size() for specialized problems

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* feat: validate overhead variable names against problem_size in find_cheapest_path

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* test: add problem_size() unit tests for all 21 problem types

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* test: add find_cheapest_path integration test with problem_size

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* fix: update test ProblemSize variable names to match problem_size() components

Existing tests used arbitrary variable names like "n" and "m" that don't
match the overhead polynomial variable names. Updated to use correct names
(num_vertices, num_edges) or empty ProblemSize to pass the new validation.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* chore: format

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* refactor: split ProblemSize into type-level names + instance-level values

Split the `Problem::problem_size()` method into two parts:
- `problem_size_names() -> &'static [&'static str]` (type-level, static)
- `problem_size_values(&self) -> Vec<usize>` (instance-level)

A free function `problem_size()` combines them into a `ProblemSize`.

This enables compile-time overhead validation: `ReductionGraph::new()`
now asserts that each reduction's overhead input variables are a subset
of the source's `problem_size_names`, and output fields are a subset of
the target's. This caught and fixed several latent bugs:

- `num_elements` → `universe_size` in SetPacking/SetCovering overheads
- `num_gates` → `num_variables`/`num_assignments` in Factoring→CircuitSAT
- `num_variables` → `num_vars` in SAT→CircuitSAT input
- `num_colors` removed from KColoring size (k is a type parameter)
- KColoring→ILP/QUBO overheads updated to not reference num_colors

Also includes earlier work from this branch:
- BicliqueCover refactored to use BipartiteGraph as input
- Polynomial composition (mul, pow, substitute) for symbolic overhead
- evaluate_path_overhead + compose_path_overhead on ReductionGraph
- 3-SAT→MIS triangular overhead test with numeric and symbolic asserts

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* refactor: extract validate_overhead_variables and use ILP for factoring test

- Extract overhead validation logic from ReductionGraph::new() and
  find_cheapest_path() into standalone validate_overhead_variables()
- Add 5 unit tests for the validation function
- Replace brute-force SpinGlass solve with ILP solver in
  test_jl_parity_factoring_to_spinglass_path (>60s → <1s)

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* fix: assert_eq in ProblemSize + correct SAT→KColoring edge overhead

- Use assert_eq! instead of debug_assert_eq! in from_names_values()
  so length mismatches are caught in release builds
- Fix SAT→KColoring num_edges polynomial: 3*num_vars + 11*num_literals
  - 9*num_clauses + 3 (was incorrectly copied from num_vertices)

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

* docs: add reduction overhead section to getting-started

Show evaluate_path_overhead and compose_path_overhead in the chained
reduction example. The getting-started guide now explains how to inspect
problem size growth along a reduction path.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 4.6 <noreply@anthropic.com>

Author: GiggleLiu

benches/solver_benchmarks.rs

@@ -197,8 +197,10 @@ fn bench_comparison(c: &mut Criterion) {
     let solver = BruteForce::new();
 
     // MaximumIndependentSet with 8 vertices
-    let is_problem =
-        MaximumIndependentSet::new(SimpleGraph::new(8, vec![(0, 1), (2, 3), (4, 5), (6, 7)]), vec![1i32; 8]);
+    let is_problem = MaximumIndependentSet::new(
+        SimpleGraph::new(8, vec![(0, 1), (2, 3), (4, 5), (6, 7)]),
+        vec![1i32; 8],
+    );
     group.bench_function("MaximumIndependentSet", |b| {
         b.iter(|| solver.find_best(black_box(&is_problem)))
     });
@@ -228,7 +230,10 @@ fn bench_comparison(c: &mut Criterion) {
     });
 
     // MaxCut with 8 vertices
-    let mc_problem = MaxCut::new(SimpleGraph::new(8, vec![(0, 1), (2, 3), (4, 5), (6, 7)]), vec![1, 1, 1, 1]);
+    let mc_problem = MaxCut::new(
+        SimpleGraph::new(8, vec![(0, 1), (2, 3), (4, 5), (6, 7)]),
+        vec![1, 1, 1, 1],
+    );
     group.bench_function("MaxCut", |b| {
         b.iter(|| solver.find_best(black_box(&mc_problem)))
     });

docs/paper/reductions.typ

@@ -246,7 +246,7 @@
 ]
 
 #block(width: 100%, inset: (x: 2em, y: 1em))[
-  *Abstract.* We present formal definitions for computational problems and polynomial-time reductions implemented in the `problemreductions` library. For each reduction, we state theorems with constructive proofs that preserve solution structure.
+  *Abstract.* We present formal definitions for computational problems and polynomial-time reductions implemented in the `problem-reductions` library. For each reduction, we state theorems with constructive proofs that preserve solution structure.
 ]