problemreductions/rules/
minimumgraphbandwidth_ilp.rs

1//! Reduction from MinimumGraphBandwidth to ILP (Integer Linear Programming).
2//!
3//! Position-assignment formulation with bandwidth variable:
4//! - Binary x_{v,p}: vertex v gets position p
5//! - Integer position variables pos_v = sum_p p * x_{v,p}
6//! - Integer bandwidth variable B
7//! - For each edge (u,v): pos_u - pos_v <= B, pos_v - pos_u <= B
8//! - Objective: minimize B
9
10use crate::models::algebraic::{LinearConstraint, ObjectiveSense, ILP};
11use crate::models::graph::MinimumGraphBandwidth;
12use crate::reduction;
13use crate::rules::traits::{ReduceTo, ReductionResult};
14use crate::topology::{Graph, SimpleGraph};
15
16/// Result of reducing MinimumGraphBandwidth to ILP.
17///
18/// Variable layout (`ILP<i64>`, non-negative integers):
19/// - `x_{v,p}` at index `v * n + p`, bounded to {0,1}
20/// - `pos_v` at index `n^2 + v`, integer position in {0, ..., n-1}
21/// - `B` (bandwidth) at index `n^2 + n`
22#[derive(Debug, Clone)]
23pub struct ReductionMGBToILP {
24    target: ILP<i64>,
25    num_vertices: usize,
26}
27
28impl ReductionResult for ReductionMGBToILP {
29    type Source = MinimumGraphBandwidth<SimpleGraph>;
30    type Target = ILP<i64>;
31
32    fn target_problem(&self) -> &ILP<i64> {
33        &self.target
34    }
35
36    /// Extract: for each vertex v, output its position p (the unique p with x_{v,p} = 1).
37    fn extract_solution(
38        &self,
39        target_solution: &<Self::Target as crate::traits::Problem>::Solution,
40    ) -> crate::rules::ExtractionResult<<Self::Source as crate::traits::Problem>::Solution> {
41        crate::rules::traits::validate_target_solution(self.target_problem(), target_solution)?;
42
43        crate::rules::ilp_helpers::one_hot_decode_rows(
44            target_solution,
45            self.num_vertices,
46            self.num_vertices,
47            0,
48        )
49    }
50}
51
52#[reduction(
53    transform = exact {
54        num_vars = "num_vertices^2 + num_vertices + 1",
55        num_constraints = "2 * num_vertices + num_vertices^2 + num_vertices + num_vertices + 1 + 2 * num_edges",
56    },
57    unavailable = {
58        num_nonzeros = "the exact target parameter is not represented by this reduction's symbolic transform",
59    }
60)]
61impl ReduceTo<ILP<i64>> for MinimumGraphBandwidth<SimpleGraph> {
62    type Result = ReductionMGBToILP;
63
64    fn reduce_to(&self) -> Result<Self::Result, crate::rules::ReductionError> {
65        let n = self.num_vertices();
66        let graph = self.graph();
67        let edges = graph.edges();
68
69        let num_x = n * n;
70        let num_vars = num_x + n + 1;
71
72        let x_idx = |v: usize, p: usize| -> usize { v * n + p };
73        let pos_idx = |v: usize| -> usize { num_x + v };
74        let b_idx = num_x + n;
75
76        let mut constraints = Vec::new();
77        let n_i64 = Self::exact_i64(n, "encoding a vertex position")?;
78
79        // Assignment: each vertex in exactly one position
80        for v in 0..n {
81            let terms: Vec<(usize, i64)> = (0..n).map(|p| (x_idx(v, p), 1)).collect();
82            constraints.push(LinearConstraint::eq(terms, 1));
83        }
84
85        // Assignment: each position has exactly one vertex
86        for p in 0..n {
87            let terms: Vec<(usize, i64)> = (0..n).map(|v| (x_idx(v, p), 1)).collect();
88            constraints.push(LinearConstraint::eq(terms, 1));
89        }
90
91        // Binary bounds for x variables (`ILP<i64>`)
92        for v in 0..n {
93            for p in 0..n {
94                constraints.push(LinearConstraint::le(vec![(x_idx(v, p), 1)], 1));
95            }
96        }
97
98        // Position variable linking: pos_v = sum_p p * x_{v,p}
99        for v in 0..n {
100            let mut terms: Vec<(usize, i64)> = vec![(pos_idx(v), 1)];
101            for p in 0..n {
102                terms.push((
103                    x_idx(v, p),
104                    -Self::exact_i64(p, "encoding a vertex position")?,
105                ));
106            }
107            constraints.push(LinearConstraint::eq(terms, 0));
108        }
109
110        // Position bounds: 0 <= pos_v <= n-1
111        for v in 0..n {
112            constraints.push(LinearConstraint::le(vec![(pos_idx(v), 1)], n_i64 - 1));
113        }
114
115        // Bandwidth upper bound: B <= n-1 (max possible position difference)
116        constraints.push(LinearConstraint::le(vec![(b_idx, 1)], n_i64 - 1));
117
118        // Bandwidth constraints: for each edge (u,v):
119        //   pos_u - pos_v <= B  =>  pos_u - pos_v - B <= 0
120        //   pos_v - pos_u <= B  =>  pos_v - pos_u - B <= 0
121        for &(u, v) in edges.iter() {
122            constraints.push(LinearConstraint::le(
123                vec![(pos_idx(u), 1), (pos_idx(v), -1), (b_idx, -1)],
124                0,
125            ));
126            constraints.push(LinearConstraint::le(
127                vec![(pos_idx(v), 1), (pos_idx(u), -1), (b_idx, -1)],
128                0,
129            ));
130        }
131
132        // Objective: minimize B
133        let objective = vec![(b_idx, 1)];
134        let target = ILP::new(num_vars, constraints, objective, ObjectiveSense::Minimize)
135            .map_err(Self::target_construction)?;
136
137        Ok(ReductionMGBToILP {
138            target,
139            num_vertices: n,
140        })
141    }
142}
143
144#[cfg(feature = "example-db")]
145pub(crate) fn canonical_rule_example_specs() -> Vec<crate::example_db::specs::RuleExampleSpec> {
146    vec![crate::example_db::specs::RuleExampleSpec {
147        id: "minimumgraphbandwidth_to_ilp",
148        build: || {
149            // Star S4: center 0 connected to 1, 2, 3
150            let source =
151                MinimumGraphBandwidth::new(SimpleGraph::new(4, vec![(0, 1), (0, 2), (0, 3)]));
152            crate::example_db::specs::rule_example_via_ilp::<_, i64>(source)
153        },
154    }]
155}
156
157#[cfg(test)]
158#[path = "../unit_tests/rules/minimumgraphbandwidth_ilp.rs"]
159mod tests;
problemreductions/rules/minimumgraphbandwidth_ilp.rs

problemreductions/rules/
minimumgraphbandwidth_ilp.rs
