Case Studies

These are synthetic cases for engineering judgment, not claims about measured production incidents.

Build graph reports every blocked package as cyclic

A dependency cycle prevents downstream packages from being emitted by Kahn's algorithm. Separate SCC cycle members from downstream blocked tasks, preserve edge direction, and return a concrete cycle witness. Condensing the cycle is useful for diagnosis but does not make its tasks executable. Evidence: partition, cycle edge IDs, blocked reachability and no false complete schedule.

Route cache accepts signed costs

A service originally accepted nonnegative route costs, then a discount edge became negative. Its Dijkstra finalization proof no longer applies. Validate the domain or select a signed-weight algorithm and classify affected targets precisely. Evidence: a negative-edge counterexample, a finite unaffected branch and a disconnected negative cycle. Do not report every route as unbounded.

Cheapest wiring mistaken for resilient wiring

An MST minimizes total connection cost but has no spare tree edges after any one-edge failure. A fault-tolerance requirement changes feasible networks. Evidence: remove one selected edge and show disconnection; revise the objective and constraints rather than promising reliability from a low weight alone.

Capacity dashboard trusts a number

A solver displays an optimum value but associates flows with the wrong parallel channels. Verify original-edge capacities, conservation, terminal net values and equal-cut capacity. Evidence: a deliberately corrupted flow vector whose displayed value is unchanged, rejected by the independent verifier.

Worker assignment stops at a maximal choice

An early greedy match blocks a worker with only one option. Alternating residual paths permit reassignment and recover maximum cardinality. Evidence: the small three-pair fixture and a matching/flow correspondence proof. If the product asks for preference stability or minimum cost, write a new contract first.