feat: Add metadata files for 12 backtracking and grid/string problems

- Add TOML metadata files for combination sum variants (39, 40, 216)
- Add TOML metadata files for permutation variants (46, 47)
- Add TOML metadata files for subset variants (78, 90)
- Add TOML metadata files for combinations (77)
- Add TOML metadata files for N-Queens II (52)
- Add TOML metadata files for word search (79)
- Add TOML metadata files for string segmentation problems (93, 131)
- All files follow ONTOLOGY_DESIGN.md specification
- Include solution-level metadata with relationships and variants

Author: lufftw

meta/problems/0039_combination_sum.toml

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+# Problem: Combination Sum
+# https://leetcode.com/problems/combination-sum/
+
+# ===== Problem Info =====
+id = "0039"
+slug = "0039_combination_sum"
+title = "Combination Sum"
+leetcode_id = 39
+url = "https://leetcode.com/problems/combination-sum/"
+
+# ===== LeetCode Official Metadata =====
+difficulty = "medium"
+topics = ["array", "backtracking"]
+companies = ["google", "amazon", "meta", "microsoft", "uber", "airbnb"]
+
+# ===== Roadmaps =====
+roadmaps = ["neetcode_150", "blind_75"]
+
+# ===== Ontology Tags (Problem Level) =====
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_combination"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0040", "0077", "0216", "0078"]
+
+# ===== File Locations =====
+[files]
+solution  = "solutions/0039_combination_sum.py"
+generator = "generators/0039_combination_sum.py"
+tests_dir = "tests/0039_combination_sum/"
+
+# ===== Solutions =====
+[[solutions]]
+key    = "default"
+class  = "Solution"
+method = "combinationSum"
+
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_combination"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0040", "0077"]
+
+role       = "base"
+variant    = "target_search_with_reuse"
+based_on   = []
+delta      = ""
+complexity = "O(n^(t/m)) time, O(t/m) space"
+notes      = "Backtracking with element reuse. Allow reuse by recursing with same index i (not i+1). Prune when candidate > remaining target."
+

meta/problems/0040_combination_sum_ii.toml

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+# Problem: Combination Sum II
+# https://leetcode.com/problems/combination-sum-ii/
+
+# ===== Problem Info =====
+id = "0040"
+slug = "0040_combination_sum_ii"
+title = "Combination Sum II"
+leetcode_id = 40
+url = "https://leetcode.com/problems/combination-sum-ii/"
+
+# ===== LeetCode Official Metadata =====
+difficulty = "medium"
+topics = ["array", "backtracking"]
+companies = ["google", "amazon", "meta", "microsoft"]
+
+# ===== Roadmaps =====
+roadmaps = ["neetcode_150"]
+
+# ===== Ontology Tags (Problem Level) =====
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_combination"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0039", "0090", "0047", "0216"]
+
+# ===== File Locations =====
+[files]
+solution  = "solutions/0040_combination_sum_ii.py"
+generator = "generators/0040_combination_sum_ii.py"
+tests_dir = "tests/0040_combination_sum_ii/"
+
+# ===== Solutions =====
+[[solutions]]
+key    = "default"
+class  = "Solution"
+method = "combinationSum2"
+
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_combination"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0039", "0090"]
+
+role       = "variant"
+variant    = "no_reuse_with_deduplication"
+based_on   = ["0039#default"]
+delta      = "No reuse: recurse with i+1. Handle duplicates: sort + same-level skip (i > start_index and candidates[i] == candidates[i-1])."
+complexity = "O(2^n) time, O(n) space"
+notes      = "Combines no-reuse pattern with same-level deduplication for handling duplicate candidates."
+

meta/problems/0046_permutations.toml

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+# Problem: Permutations
+# https://leetcode.com/problems/permutations/
+
+# ===== Problem Info =====
+id = "0046"
+slug = "0046_permutations"
+title = "Permutations"
+leetcode_id = 46
+url = "https://leetcode.com/problems/permutations/"
+
+# ===== LeetCode Official Metadata =====
+difficulty = "medium"
+topics = ["array", "backtracking"]
+companies = ["google", "amazon", "meta", "microsoft", "apple", "uber"]
+
+# ===== Roadmaps =====
+roadmaps = ["neetcode_150"]
+
+# ===== Ontology Tags (Problem Level) =====
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_permutation"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0047", "0078", "0077"]
+
+# ===== Pattern Role =====
+[pattern_role]
+is_base_template = true
+base_for_kernel = "BacktrackingExploration"
+derived_problems = ["0047"]
+
+# ===== File Locations =====
+[files]
+solution  = "solutions/0046_permutations.py"
+generator = "generators/0046_permutations.py"
+tests_dir = "tests/0046_permutations/"
+
+# ===== Solutions =====
+[[solutions]]
+key    = "default"
+class  = "Solution"
+method = "permute"
+
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_permutation"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0047"]
+
+role       = "base"
+variant    = ""
+based_on   = []
+delta      = ""
+complexity = "O(n! × n) time, O(n) space"
+notes      = "BASE TEMPLATE for permutation enumeration. Track usage with boolean array. At each position, try every unused element. This is the canonical backtracking permutation pattern."
+

meta/problems/0047_permutations_ii.toml

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+# Problem: Permutations II
+# https://leetcode.com/problems/permutations-ii/
+
+# ===== Problem Info =====
+id = "0047"
+slug = "0047_permutations_ii"
+title = "Permutations II"
+leetcode_id = 47
+url = "https://leetcode.com/problems/permutations-ii/"
+
+# ===== LeetCode Official Metadata =====
+difficulty = "medium"
+topics = ["array", "backtracking"]
+companies = ["google", "amazon", "meta", "microsoft"]
+
+# ===== Roadmaps =====
+roadmaps = []
+
+# ===== Ontology Tags (Problem Level) =====
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_permutation"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0046", "0040", "0090"]
+
+# ===== File Locations =====
+[files]
+solution  = "solutions/0047_permutations_ii.py"
+generator = "generators/0047_permutations_ii.py"
+tests_dir = "tests/0047_permutations_ii/"
+
+# ===== Solutions =====
+[[solutions]]
+key    = "default"
+class  = "Solution"
+method = "permuteUnique"
+
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_permutation"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0046"]
+
+role       = "variant"
+variant    = "with_duplicates"
+based_on   = ["0046#default"]
+delta      = "Sort array + same-level deduplication: skip nums[i] if i > 0 and nums[i] == nums[i-1] and !used[i-1]. Ensures leftmost duplicate is always picked first at each level."
+complexity = "O(n! × n) time, O(n) space"
+notes      = "Extension of base permutation pattern with duplicate handling using same-level deduplication."
+

meta/problems/0052_n_queens_ii.toml

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+# Problem: N-Queens II
+# https://leetcode.com/problems/n-queens-ii/
+
+# ===== Problem Info =====
+id = "0052"
+slug = "0052_n_queens_ii"
+title = "N-Queens II"
+leetcode_id = 52
+url = "https://leetcode.com/problems/n-queens-ii/"
+
+# ===== LeetCode Official Metadata =====
+difficulty = "hard"
+topics = ["backtracking"]
+companies = ["google", "amazon", "meta"]
+
+# ===== Roadmaps =====
+roadmaps = []
+
+# ===== Ontology Tags (Problem Level) =====
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_n_queens"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array", "hash_set"]
+algorithms       = ["backtracking"]
+related_problems = ["0051"]
+
+# ===== File Locations =====
+[files]
+solution  = "solutions/0052_n_queens_ii.py"
+generator = "generators/0052_n_queens_ii.py"
+tests_dir = "tests/0052_n_queens_ii/"
+
+# ===== Solutions =====
+[[solutions]]
+key    = "default"
+class  = "SolutionSets"
+method = "totalNQueens"
+
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_n_queens"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array", "hash_set"]
+algorithms       = ["backtracking"]
+related_problems = ["0051"]
+
+role       = "variant"
+variant    = "count_only"
+based_on   = ["0051#default"]
+delta      = "Only count solutions instead of building board representations. More memory efficient."
+complexity = "O(n!) time, O(n) space"
+notes      = "Same algorithm as N-Queens but optimized for counting. Uses hash sets for O(1) constraint checking."
+
+[[solutions]]
+key    = "bitmask"
+class  = "SolutionBitmask"
+method = "totalNQueens"
+
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_n_queens"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0051"]
+
+role       = "variant"
+variant    = "bitmask_optimization"
+based_on   = ["0052#default"]
+delta      = "Use integers as bitmasks for ultra-fast constraint checking. Bitwise operations are faster than hash lookups."
+complexity = "O(n!) time, O(n) space"
+notes      = "Optimized version using bitmasks for constraint tracking. Better cache locality and smaller constants."
+

meta/problems/0077_combinations.toml

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+# Problem: Combinations
+# https://leetcode.com/problems/combinations/
+
+# ===== Problem Info =====
+id = "0077"
+slug = "0077_combinations"
+title = "Combinations"
+leetcode_id = 77
+url = "https://leetcode.com/problems/combinations/"
+
+# ===== LeetCode Official Metadata =====
+difficulty = "medium"
+topics = ["backtracking"]
+companies = ["google", "amazon", "meta"]
+
+# ===== Roadmaps =====
+roadmaps = []
+
+# ===== Ontology Tags (Problem Level) =====
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_combination"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0078", "0039", "0216"]
+
+# ===== File Locations =====
+[files]
+solution  = "solutions/0077_combinations.py"
+generator = "generators/0077_combinations.py"
+tests_dir = "tests/0077_combinations/"
+
+# ===== Solutions =====
+[[solutions]]
+key    = "default"
+class  = "Solution"
+method = "combine"
+
+api_kernels      = ["BacktrackingExploration"]
+patterns         = ["backtracking_combination"]
+families         = ["backtracking_combinatorial"]
+data_structures  = ["array"]
+algorithms       = ["backtracking"]
+related_problems = ["0078", "0039"]
+
+role       = "variant"
+variant    = "fixed_size_subset"
+based_on   = ["0078#default"]
+delta      = "Fixed-size subset enumeration. Only collect when path length equals k. Add count-based pruning: stop early if not enough elements remain."
+complexity = "O(k × C(n,k)) time, O(k) space"
+notes      = "Similar to subsets but with fixed size constraint. Uses start_index for canonical ordering and pruning optimization."
+