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Add greedy set cover approximation algorithm
Describe your change:
This PR implements the greedy approximation algorithm for the minimum set cover problem, a fundamental NP-complete problem in computer science with applications in resource allocation, scheduling, and combinatorial optimization.
Implementation highlights:
- Standard greedy heuristic with $H_d$ approximation guarantee ($\leq \ln|U| + 1$)
- Comprehensive input validation and error handling
- Full doctest coverage (17 tests) including edge cases
- Detailed theoretical documentation with references
- Time complexity: $O(|U| × |S|)$, Space complexity: $O(|U| + |S|)$
- [x] Add an algorithm?
- [ ] Fix a bug or typo in an existing algorithm?
- [ ] Add or change doctests? -- Note: Please avoid changing both code and tests in a single pull request.
- [ ] Documentation change?
Checklist:
- [x] I have read CONTRIBUTING.md.
- [x] This pull request is all my own work -- I have not plagiarized.
- [x] I know that pull requests will not be merged if they fail the automated tests.
- [x] This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
- [x] All new Python files are placed inside an existing directory.
- [x] All filenames are in all lowercase characters with no spaces or dashes.
- [x] All functions and variable names follow Python naming conventions.
- [x] All function parameters and return values are annotated with Python type hints.
- [x] All functions have doctests that pass the automated testing.
- [x] All new algorithms include at least one URL that points to Wikipedia or another similar explanation.
- [ ] If this pull request resolves one or more open issues then the description above includes the issue number(s) with a closing keyword: "Fixes #ISSUE-NUMBER".
References:
- Wikipedia: Set Cover Problem
- Vazirani, Vijay V. Approximation Algorithms. Berlin: Springer, 2001.
- Karp, R. M. (1972). "Reducibility Among Combinatorial Problems"
- Johnson, D. S. (1974). "Approximation algorithms for combinatorial problems"
- Feige, U. (1998). "A Threshold of ln n for Approximating Set Cover", Journal of the ACM, 45(4), 634-652
- Cormen et al., "Introduction to Algorithms", Chapter 35.3: The Set-Covering Problem
Testing:
All automated tests pass locally:
python -m doctest -v greedy_methods/greedy_set_cover.py
ruff check greedy_methods/greedy_set_cover.py
black --check greedy_methods/greedy_set_cover.py
mypy --ignore-missing-imports greedy_methods/greedy_set_cover.py