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Paths, Proofs, and Perfection: Developing a Human-Interpretable Proof System for Constrained Shortest Paths
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Proposes a graph-aware proof system and A*-like proof search that produce human-auditable optimality certificates for constrained shortest paths, yielding smaller proofs than an integer programming certificate baseline on road-network instances.
Problem
Certifying optimality of constrained shortest-path solutions is hard for non-experts; general-purpose certificates (MIP cutting planes, SAT/PB proofs) are large and lack graph-level intuition.
Approach
A proof system over intervals of graphs [G_bottom, G_top] with graph-theoretic reasoning rules (e.g., out-of-range vertex removal, mandatory vertices from connectivity) plus case branching on vertices/edges, and an A*-style proof search that minimizes the number of leaves using a bounding function.
Key results
- Pure bounding variant produces shorter proofs than the MIP baseline in 72.8% of feasible mandatory-vertex instances.
- Success rates: mandatory-vertex 96.29% (Pure), resource 92.40% (Pure) for constructing proofs within the time limit.
- For resource-constraint instances the proposed approach does not improve proof size, since the baseline solves most (>95%) with at most one branching level.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the pure bound rather than the Lex or Weighted side-constraint-aware bounds (ablation) | Proof search for constrained shortest paths · Mandatory-vertex instances | proof construction success rate: 96.29% vs. 93.57%; 96.29% vs. 95.93% | Helped |
| Use the pure bound rather than the Lex or Weighted side-constraint-aware bounds (ablation) | Proof search for constrained shortest paths · Resource-constraint instances | proof construction success rate: 92.40% vs. 42.88%; 92.40% vs. 72.93% | Helped |
| Use the proposed proof system instead of the integer-programming baseline | Proof generation · Feasible mandatory-vertex instances | proof width: 72.8% of cases | Helped |
| Use the proposed proof system instead of the integer-programming baseline | Proof generation · Infeasible mandatory-vertex instances | proof width: 22% cases | Mixed |
| Use the proposed proof system instead of the integer-programming baseline | Proof generation · Resource-constraint instances | proof width | Hurt |
| Use the proposed proof system instead of the integer-programming baseline | Proof generation · Feasible resource-constraint instances | proof width: > 95% | Hurt |
Surprising findings
- Using side-constraint information in the bounding function (Lex, Weighted) reduced success rates, especially for resource constraints (Lex 42.88%).
Limitations
- Proof search does not guarantee the smallest proof; it stops at the first valid proof.
- Resource-constraint instances show no proof-size improvement over the MIP baseline.
- Side constraints are treated as a black box, and the bounding heuristics that use them hurt success rates.
Experimental setup
Datasets: OpenStreetMap road networks
Benchmarks: constrained shortest path instances from OpenStreetMap road networks with mandatory vertex and resource constraints
Metrics: proof width (number of leaves) success rate proof depth ratio to baseline
Baselines: integer programming with subtour elimination and MIP certificate (Cheung, Gleixn
Builds on: cutting-plane proof system branch-and-cut certificates (Cheung et al. 2017) Dijkstra's algorithm A* search QuickXplain
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