Optimal single-machine job sequence (weighted tardiness with precedence)

# Optimal job sequence: minimum total weighted tardiness with precedence `input.json` (synthetic) has `start` (time the machine becomes free), `jobs` (`id`, processing time `p`, weight `w`, `due`) and `precedence` (pairs [a, b]: a must come before b, not necessarily directly). One machine, no idle time, no preemption. Write `solve.py` (Python 3 standard library only); `python solve.py input.json answer.json`. Find a sequence of all jobs that respects every precedence pair and **minimises** total weighted tardiness: sum of w * max(0, C - due), where C is the job's completion time (start + sum of p up to and including the job). Any optimal sequence is accepted. Fresh instances have **20 and 21 jobs**; each run has a 90-second limit. ## answer.json `{"sequence": [job ids in order], "total_weighted_tardiness": integer}` ## Scoring Hidden checks (partial credit; all must pass). A sequence is **valid** if it is a permutation of the job ids, respects every precedence pair, and the reported objective equals the scorer's recomputation for that sequence. It is **optimal** if that objective equals the true minimum (the scorer has the optimum). `fixture_valid_sequence`, `fixture_optimal` (given 18-job instance); `fresh_20_jobs_valid`, `fresh_20_jobs_optimal`; `fresh_21_jobs_optimal`.