Core Concepts
An efficient polynomial-time algorithm for enumerating all extreme points of a bisubmodular polyhedron.
Abstract
The content presents an efficient algorithm for enumerating all the extreme points of a bisubmodular polyhedron. The key highlights are:
The algorithm uses the reverse search technique proposed by Avis and Fukuda to avoid redundant searches.
It leverages the characterization of the adjacency of extreme points using signed posets, as developed by Ando and Fujishige.
The algorithm has a time complexity of O(n^4|V|) and a space complexity of O(n^2), where n is the dimension of the underlying space and |V| is the number of extreme points.
The algorithm also has an O(n^6) time delay, meaning the time between two consecutive outputs is bounded by a polynomial function of the input size.
The capacity functions required in the algorithm can be computed efficiently in constant time, without the need to solve bisubmodular function minimization problems.
The algorithm is a generalization of the enumeration algorithm for base polyhedra, which encompasses various combinatorial enumeration problems.
The algorithm has potential applications in machine learning and artificial intelligence, where bisubmodular functions are used to model deep multivariate submodular functions.