Create fractional_cover_problem.py (#9973)

* Create fractional_cover_problem.py

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* Update fractional_cover_problem.py

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* Update fractional_cover_problem.py

* Update fractional_cover_problem.py

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* Update fractional_cover_problem.py

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* Update fractional_cover_problem.py

* Define Item property ratio

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

greedy_methods/fractional_cover_problem.py

@@ -0,0 +1,102 @@
+# https://en.wikipedia.org/wiki/Set_cover_problem
+
+from dataclasses import dataclass
+from operator import attrgetter
+
+
+@dataclass
+class Item:
+    weight: int
+    value: int
+
+    @property
+    def ratio(self) -> float:
+        """
+        Return the value-to-weight ratio for the item.
+
+        Returns:
+            float: The value-to-weight ratio for the item.
+
+        Examples:
+        >>> Item(10, 65).ratio
+        6.5
+
+        >>> Item(20, 100).ratio
+        5.0
+
+        >>> Item(30, 120).ratio
+        4.0
+        """
+        return self.value / self.weight
+
+
+def fractional_cover(items: list[Item], capacity: int) -> float:
+    """
+    Solve the Fractional Cover Problem.
+
+    Args:
+        items: A list of items, where each item has weight and value attributes.
+        capacity: The maximum weight capacity of the knapsack.
+
+    Returns:
+        The maximum value that can be obtained by selecting fractions of items to cover
+        the knapsack's capacity.
+
+    Raises:
+        ValueError: If capacity is negative.
+
+    Examples:
+    >>> fractional_cover((Item(10, 60), Item(20, 100), Item(30, 120)), capacity=50)
+    240.0
+
+    >>> fractional_cover([Item(20, 100), Item(30, 120), Item(10, 60)], capacity=25)
+    135.0
+
+    >>> fractional_cover([Item(10, 60), Item(20, 100), Item(30, 120)], capacity=60)
+    280.0
+
+    >>> fractional_cover(items=[Item(5, 30), Item(10, 60), Item(15, 90)], capacity=30)
+    180.0
+
+    >>> fractional_cover(items=[], capacity=50)
+    0.0
+
+    >>> fractional_cover(items=[Item(10, 60)], capacity=5)
+    30.0
+
+    >>> fractional_cover(items=[Item(10, 60)], capacity=1)
+    6.0
+
+    >>> fractional_cover(items=[Item(10, 60)], capacity=0)
+    0.0
+
+    >>> fractional_cover(items=[Item(10, 60)], capacity=-1)
+    Traceback (most recent call last):
+        ...
+    ValueError: Capacity cannot be negative
+    """
+    if capacity < 0:
+        raise ValueError("Capacity cannot be negative")
+
+    total_value = 0.0
+    remaining_capacity = capacity
+
+    # Sort the items by their value-to-weight ratio in descending order
+    for item in sorted(items, key=attrgetter("ratio"), reverse=True):
+        if remaining_capacity == 0:
+            break
+
+        weight_taken = min(item.weight, remaining_capacity)
+        total_value += weight_taken * item.ratio
+        remaining_capacity -= weight_taken
+
+    return total_value
+
+
+if __name__ == "__main__":
+    import doctest
+
+    if result := doctest.testmod().failed:
+        print(f"{result} test(s) failed")
+    else:
+        print("All tests passed")