The Volume of a Crosspolytope Truncated by a Halfspace

• Published in: Theory and Applications of Models of Computation Vol. 11436; pp. 13 - 27
• Main Authors: Ando, Ei, Tsuchiya, Shoichi
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 01.01.2019; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Geometric duality; Polynomial time algorithm; Volume computation
• ISBN: 3030148114; 9783030148119
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-14812-6_2

In this paper, we consider the computation of the volume of an n-dimensional crosspolytope truncated by a halfspace. Since a crosspolytope has exponentially many facets, we cannot efficiently compute the volume by dividing the truncated crosspolytope into simplices. We show an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^6)$$\end{document} time algorithm for the computation of the volume. This makes a contrast to the 0−1 knapsack polytope, whose volume is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\#P$$\end{document}-hard to compute. The paper is interested in the computation of the volume of the truncated crosspolytope because we conjecture the following question may have an affirmative answer: Does the existence of a polynomial time algorithm for the computation of the volume of a polytope K imply the same for K’s geometric dual? We give one example where the answer is yes.