update

Author: GiggleLiu

.claude/skills/check-issue/SKILL.md

@@ -195,6 +195,7 @@ If the algorithm is a high-level sketch rather than an implementable procedure 
 - **Non-trivial**: Must have enough structure to exercise the reduction meaningfully (not just 2 vertices)
 - **Brute-force solvable**: Small enough to verify by hand or with `pred solve`
 - **Fully worked**: Shows the source instance, the reduction construction step by step, and the target instance — not just "apply the reduction to get..."
+- **Round-trip testable**: The example must be complex enough to validate correctness via a closed-loop test: reduce the source instance → solve the target → extract the solution back → verify it is optimal for the source. A too-simple example (e.g., a single edge, a trivially satisfiable formula) can pass the round trip even with a buggy reduction. The example should have multiple feasible solutions with different objective values so that only a correct reduction maps to the true optimum. Rule of thumb: the source instance should have at least 2 suboptimal feasible solutions in addition to the optimal one.
 
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@@ -329,6 +330,7 @@ The formal definition must be **precise and implementable**:
 - **Exercises core structure**: Examples must use the defining features of the problem. For instance, a "MultivariateQuadratic" example that only has linear terms does not exercise the quadratic structure → **Fail**. If the problem's name or definition highlights a specific structural feature (quadratic, k-colorable, bipartite, etc.), at least one example must exercise that feature.
 - **Known optimal solution provided**: Must state the optimal value, not just the instance
 - **Detailed enough for paper**: This example will appear in the paper — it needs to be illustrative
+- **Round-trip testable**: The example must be complex enough that a round-trip test (construct instance → solve → verify) can catch implementation bugs. A too-simple instance (e.g., 2 vertices, a single clause) may have a trivially correct solution that passes even with a wrong implementation. The example should have multiple feasible configurations with different objective values (for optimization) or a mix of satisfying and non-satisfying configurations (for satisfaction problems), so that correctness is meaningfully tested. Rule of thumb: the instance should have at least 2 suboptimal feasible solutions in addition to the optimal one.
 
 ### 4e: Representation Feasibility