n维点集凸包内最大体积单纯形的非穷举求解算法咨询

Great question! Let’s break this down with practical geometric insights and known results:

First off, a critical foundational observation: the vertices of the maximum-volume simplex inside a convex hull C must all be vertices of C itself. Any simplex using an interior point can be "expanded" by replacing that point with a convex hull vertex to get a larger (or equal) volume. So we only need to consider subsets of C’s vertex set, not all points in C.

Now, onto your core question—are there non-exhaustive algorithms for this? The answer depends on whether the dimension n is fixed or scales with the input:

Fixed Dimension (n is a constant, e.g., 2D, 3D) #

For fixed n, efficient non-exhaustive exact algorithms exist, avoiding the brute-force O(m^(n+1)) complexity (where m is the number of convex hull vertices):

• 2D (Convex Polygon → Maximum Triangle): The classic rotation calipers method runs in O(n) time. Instead of checking all triplets of vertices, it iterates over each edge of the convex hull and finds the vertex farthest from that edge, computing the triangle volume in constant time per edge. This leverages the convex polygon’s monotonicity properties to skip redundant checks.
• 3D (Convex Polyhedron → Maximum Tetrahedron): While trickier than 2D, there are O(m²) exact algorithms. These use geometric properties (e.g., the maximum tetrahedron must have at least one face lying on a face of the convex hull, or edges aligned with extreme directions) to prune the candidate vertex sets, avoiding checking all m choose 4 combinations.
• Higher Fixed n: For n ≥ 4 (fixed), polynomial-time exact algorithms exist. These often frame the problem as maximizing the determinant of a matrix formed by the simplex vertices (since simplex volume is proportional to this determinant), and use convex optimization or combinatorial geometric techniques to narrow down candidate vertex subsets without exhaustive enumeration.

Variable Dimension (n scales with input size) #

When n is not fixed (e.g., n grows with the number of convex hull vertices), the problem is NP-hard. This means no polynomial-time exact algorithm is known (unless P=NP), but there are still non-exhaustive approximate algorithms:

• Random Sampling: Randomly select subsets of n+1 convex hull vertices, compute their simplex volume, and retain the maximum found. This is non-exhaustive and can get close to the optimal solution with enough samples.
• Polynomial-Time Approximation Schemes (PTAS): These algorithms can produce a simplex with volume within (1-ε) of the maximum (for any ε>0) in polynomial time, using geometric approximations and pruning techniques to avoid full enumeration.
• Heuristic Methods: Greedy algorithms that start with a small simplex and iteratively replace vertices to increase volume, stopping when no improvement is possible. These are fast and non-exhaustive, though they don’t guarantee the global maximum.

Key Takeaway #

• For fixed low/medium dimensions (like 2D, 3D, or even 10D if m is manageable), you can use efficient non-exhaustive exact algorithms.
• For high, variable dimensions, stick to non-exhaustive approximate or heuristic methods—exact exhaustive checks are infeasible here.

内容的提问来源于stack exchange，提问作者Nico Schlömer