Fix #207: [Rule] MinimumVertexCover to MinimumFeedbackVertexSet (#713)

GiggleLiu · web-flow · commit 1d7128b9f975 · 2026-03-21T12:15:21.000+08:00
* Add plan for #207: MinimumVertexCover to MinimumFeedbackVertexSet * Implement #207: add MinimumVertexCover to MinimumFeedbackVertexSet reduction * chore: remove plan file after implementation
diff --git a/docs/paper/reductions.typ b/docs/paper/reductions.typ
@@ -4252,6 +4252,28 @@ Each reduction is presented as a *Rule* (with linked problem names and overhead
   _Solution extraction._ For IS solution $S$, return $C = V backslash S$, i.e.\ flip each variable: $c_v = 1 - s_v$.
 ]
 
+#let mvc_fvs = load-example("MinimumVertexCover", "MinimumFeedbackVertexSet")
+#let mvc_fvs_sol = mvc_fvs.solutions.at(0)
+#let mvc_fvs_cover = mvc_fvs_sol.source_config.enumerate().filter(((i, x)) => x == 1).map(((i, x)) => i)
+#let mvc_fvs_fvs = mvc_fvs_sol.target_config.enumerate().filter(((i, x)) => x == 1).map(((i, x)) => i)
+#reduction-rule("MinimumVertexCover", "MinimumFeedbackVertexSet",
+  example: true,
+  example-caption: [7-vertex graph: each source edge becomes a directed 2-cycle],
+  extra: [
+    Source VC: $C = {#mvc_fvs_cover.map(str).join(", ")}$ (size #mvc_fvs_cover.len()) on a graph with $n = #graph-num-vertices(mvc_fvs.source.instance)$ vertices and $|E| = #graph-num-edges(mvc_fvs.source.instance)$ edges \
+    Target FVS: $F = {#mvc_fvs_fvs.map(str).join(", ")}$ (size #mvc_fvs_fvs.len()) on a digraph with the same $n = #graph-num-vertices(mvc_fvs.target.instance)$ vertices and $|A| = #mvc_fvs.target.instance.graph.arcs.len() = 2 |E|$ arcs \
+    Canonical witness is preserved exactly: $C = F$ #sym.checkmark
+  ],
+)[
+  Each undirected edge $\{u, v\}$ can be viewed as the directed 2-cycle $u -> v -> u$. Replacing every source edge this way turns the task "hit every edge with a chosen endpoint" into "hit every directed cycle with a chosen vertex." The vertex set, weights, and budget are preserved, so the reduction is size-preserving up to doubling the edge count into arcs.
+][
+  _Construction._ Given a Minimum Vertex Cover instance $(G = (V, E), bold(w))$, build the directed graph $D = (V, A)$ on the same vertex set, where for every undirected edge $\{u, v\} in E$ we add both arcs $(u, v)$ and $(v, u)$ to $A$. Keep the vertex weights unchanged and reuse the same decision variables $x_v in {0,1}$.
+
+  _Correctness._ ($arrow.r.double$) If $C subset.eq V$ is a vertex cover of $G$, then every source edge $\{u, v\}$ has an endpoint in $C$, so the corresponding 2-cycle $u -> v -> u$ in $D$ is hit by $C$. Any longer directed cycle in $D$ is also made from source edges, so one of its vertices lies in $C$ as well. Therefore removing $C$ destroys all directed cycles, and $C$ is a feedback vertex set of $D$. ($arrow.l.double$) If $F subset.eq V$ is a feedback vertex set of $D$, then for every source edge $\{u, v\}$ the digraph contains the 2-cycle $u -> v -> u$, which must be hit by $F$. Hence at least one of $u, v$ lies in $F$, so $F$ covers every edge of $G$ and is a vertex cover.
+
+  _Solution extraction._ Return the target solution vector unchanged: a selected vertex in the feedback vertex set is selected in the vertex cover, and vice versa.
+]
+
 #reduction-rule("MaximumIndependentSet", "MinimumVertexCover")[
   The exact reverse of VC $arrow.r$ IS: complementing an independent set yields a vertex cover. The graph and weights are preserved unchanged, and $|"IS"| + |"VC"| = |V|$ ensures optimality carries over.
 ][
diff --git a/src/rules/minimumvertexcover_minimumfeedbackvertexset.rs b/src/rules/minimumvertexcover_minimumfeedbackvertexset.rs
@@ -0,0 +1,97 @@
+//! Reduction from MinimumVertexCover to MinimumFeedbackVertexSet.
+//!
+//! Each undirected edge becomes a directed 2-cycle, so a vertex cover is
+//! exactly a feedback vertex set in the constructed digraph.
+
+use crate::models::graph::{MinimumFeedbackVertexSet, MinimumVertexCover};
+use crate::reduction;
+use crate::rules::traits::{ReduceTo, ReductionResult};
+use crate::topology::{DirectedGraph, Graph, SimpleGraph};
+use crate::types::WeightElement;
+
+/// Result of reducing MinimumVertexCover to MinimumFeedbackVertexSet.
+#[derive(Debug, Clone)]
+pub struct ReductionVCToFVS<W> {
+    target: MinimumFeedbackVertexSet<W>,
+}
+
+impl<W> ReductionResult for ReductionVCToFVS<W>
+where
+    W: WeightElement + crate::variant::VariantParam,
+{
+    type Source = MinimumVertexCover<SimpleGraph, W>;
+    type Target = MinimumFeedbackVertexSet<W>;
+
+    fn target_problem(&self) -> &Self::Target {
+        &self.target
+    }
+
+    fn extract_solution(&self, target_solution: &[usize]) -> Vec<usize> {
+        target_solution.to_vec()
+    }
+}
+
+#[reduction(
+    overhead = {
+        num_vertices = "num_vertices",
+        num_arcs = "2 * num_edges",
+    }
+)]
+impl ReduceTo<MinimumFeedbackVertexSet<i32>> for MinimumVertexCover<SimpleGraph, i32> {
+    type Result = ReductionVCToFVS<i32>;
+
+    fn reduce_to(&self) -> Self::Result {
+        let arcs = self
+            .graph()
+            .edges()
+            .into_iter()
+            .flat_map(|(u, v)| [(u, v), (v, u)])
+            .collect();
+
+        let target = MinimumFeedbackVertexSet::new(
+            DirectedGraph::new(self.graph().num_vertices(), arcs),
+            self.weights().to_vec(),
+        );
+
+        ReductionVCToFVS { target }
+    }
+}
+
+#[cfg(feature = "example-db")]
+pub(crate) fn canonical_rule_example_specs() -> Vec<crate::example_db::specs::RuleExampleSpec> {
+    use crate::export::SolutionPair;
+
+    vec![crate::example_db::specs::RuleExampleSpec {
+        id: "minimumvertexcover_to_minimumfeedbackvertexset",
+        build: || {
+            let source = MinimumVertexCover::new(
+                SimpleGraph::new(
+                    7,
+                    vec![
+                        (0, 1),
+                        (0, 2),
+                        (0, 3),
+                        (1, 2),
+                        (1, 3),
+                        (3, 4),
+                        (4, 5),
+                        (5, 6),
+                    ],
+                ),
+                vec![1i32; 7],
+            );
+
+            crate::example_db::specs::rule_example_with_witness::<_, MinimumFeedbackVertexSet<i32>>(
+                source,
+                SolutionPair {
+                    source_config: vec![1, 1, 0, 1, 0, 1, 0],
+                    target_config: vec![1, 1, 0, 1, 0, 1, 0],
+                },
+            )
+        },
+    }]
+}
+
+#[cfg(test)]
+#[path = "../unit_tests/rules/minimumvertexcover_minimumfeedbackvertexset.rs"]
+mod tests;
diff --git a/src/rules/mod.rs b/src/rules/mod.rs
@@ -29,6 +29,7 @@ mod maximumsetpacking_casts;
 pub(crate) mod maximumsetpacking_qubo;
 pub(crate) mod minimummultiwaycut_qubo;
 pub(crate) mod minimumvertexcover_maximumindependentset;
+pub(crate) mod minimumvertexcover_minimumfeedbackvertexset;
 pub(crate) mod minimumvertexcover_minimumsetcovering;
 pub(crate) mod sat_circuitsat;
 pub(crate) mod sat_coloring;
@@ -112,6 +113,7 @@ pub(crate) fn canonical_rule_example_specs() -> Vec<crate::example_db::specs::Ru
     specs.extend(maximumsetpacking_qubo::canonical_rule_example_specs());
     specs.extend(minimummultiwaycut_qubo::canonical_rule_example_specs());
     specs.extend(minimumvertexcover_maximumindependentset::canonical_rule_example_specs());
+    specs.extend(minimumvertexcover_minimumfeedbackvertexset::canonical_rule_example_specs());
     specs.extend(minimumvertexcover_minimumsetcovering::canonical_rule_example_specs());
     specs.extend(sat_circuitsat::canonical_rule_example_specs());
     specs.extend(sat_coloring::canonical_rule_example_specs());
diff --git a/src/unit_tests/rules/minimumvertexcover_minimumfeedbackvertexset.rs b/src/unit_tests/rules/minimumvertexcover_minimumfeedbackvertexset.rs
@@ -0,0 +1,130 @@
+#[cfg(feature = "example-db")]
+use super::canonical_rule_example_specs;
+use super::ReductionVCToFVS;
+use crate::models::graph::{MinimumFeedbackVertexSet, MinimumVertexCover};
+use crate::rules::test_helpers::assert_optimization_round_trip_from_optimization_target;
+use crate::rules::traits::ReductionResult;
+use crate::rules::ReduceTo;
+#[cfg(feature = "example-db")]
+use crate::solvers::{BruteForce, Solver};
+use crate::topology::{Graph, SimpleGraph};
+#[cfg(feature = "example-db")]
+use crate::traits::Problem;
+
+fn weighted_cycle_cover_source() -> MinimumVertexCover<SimpleGraph, i32> {
+    MinimumVertexCover::new(
+        SimpleGraph::new(5, vec![(0, 1), (1, 2), (2, 0), (2, 3), (3, 4)]),
+        vec![4, 1, 3, 2, 5],
+    )
+}
+
+#[test]
+fn test_minimumvertexcover_to_minimumfeedbackvertexset_closed_loop() {
+    let source = weighted_cycle_cover_source();
+    let reduction: ReductionVCToFVS<i32> =
+        ReduceTo::<MinimumFeedbackVertexSet<i32>>::reduce_to(&source);
+
+    assert_optimization_round_trip_from_optimization_target(
+        &source,
+        &reduction,
+        "MVC -> FVS closed loop",
+    );
+}
+
+#[test]
+fn test_reduction_structure() {
+    let source = weighted_cycle_cover_source();
+    let reduction: ReductionVCToFVS<i32> =
+        ReduceTo::<MinimumFeedbackVertexSet<i32>>::reduce_to(&source);
+    let target = reduction.target_problem();
+
+    assert_eq!(target.graph().num_vertices(), source.graph().num_vertices());
+    assert_eq!(target.num_arcs(), 2 * source.num_edges());
+
+    let mut arcs = target.graph().arcs();
+    arcs.sort_unstable();
+
+    assert_eq!(
+        arcs,
+        vec![
+            (0, 1),
+            (0, 2),
+            (1, 0),
+            (1, 2),
+            (2, 0),
+            (2, 1),
+            (2, 3),
+            (3, 2),
+            (3, 4),
+            (4, 3),
+        ]
+    );
+}
+
+#[test]
+fn test_weight_preservation() {
+    let source = weighted_cycle_cover_source();
+    let reduction: ReductionVCToFVS<i32> =
+        ReduceTo::<MinimumFeedbackVertexSet<i32>>::reduce_to(&source);
+
+    assert_eq!(reduction.target_problem().weights(), source.weights());
+}
+
+#[test]
+fn test_identity_solution_extraction() {
+    let source = weighted_cycle_cover_source();
+    let reduction: ReductionVCToFVS<i32> =
+        ReduceTo::<MinimumFeedbackVertexSet<i32>>::reduce_to(&source);
+
+    assert_eq!(
+        reduction.extract_solution(&[1, 0, 1, 0, 1]),
+        vec![1, 0, 1, 0, 1]
+    );
+}
+
+#[cfg(feature = "example-db")]
+#[test]
+fn test_canonical_rule_example_spec_builds() {
+    let example = (canonical_rule_example_specs()
+        .into_iter()
+        .find(|spec| spec.id == "minimumvertexcover_to_minimumfeedbackvertexset")
+        .expect("example spec should be registered")
+        .build)();
+
+    assert_eq!(example.source.problem, "MinimumVertexCover");
+    assert_eq!(example.target.problem, "MinimumFeedbackVertexSet");
+    assert_eq!(example.solutions.len(), 1);
+    assert_eq!(
+        example.solutions[0].source_config,
+        example.solutions[0].target_config
+    );
+
+    let source: MinimumVertexCover<SimpleGraph, i32> =
+        serde_json::from_value(example.source.instance.clone())
+            .expect("source example deserializes");
+    let target: MinimumFeedbackVertexSet<i32> =
+        serde_json::from_value(example.target.instance.clone())
+            .expect("target example deserializes");
+    let solution = &example.solutions[0];
+
+    let source_metric = source.evaluate(&solution.source_config);
+    let target_metric = target.evaluate(&solution.target_config);
+    assert!(
+        source_metric.is_valid(),
+        "source witness should be feasible"
+    );
+    assert!(
+        target_metric.is_valid(),
+        "target witness should be feasible"
+    );
+
+    let best_source = BruteForce::new()
+        .find_best(&source)
+        .expect("source example should have an optimum");
+    let best_target = BruteForce::new()
+        .find_best(&target)
+        .expect("target example should have an optimum");
+
+    assert_eq!(source_metric, source.evaluate(&best_source));
+    assert_eq!(target_metric, target.evaluate(&best_target));
+}
