Unmixing the Mixed Volume Computation

Abstract

Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized volume of the convex hull of their union. Under these conditions the problem of computing mixed volume of several polytopes can be transformed into a volume computation problem for a single polytope in the same dimension. We demonstrate through problems from real world applications that substantial reduction in computational costs can be achieved via this transformation in situations where the convex hull of the union of the polytopes has less complex geometry than the original polytopes. We also discuss the important implications of this result in the polyhedral homotopy method for solving polynomial systems.

Appendices

A: Monotonicity of Mixed Volume

The mixed volume ${{\,\mathrm{MV}\,}}(Q_1,\dots ,Q_n)$, as a function that takes n convex polytopes, is monotone in each of its arguments in the sense that if $Q_1' \subseteq Q_1$ then ${{\,\mathrm{MV}\,}}(Q_1',Q_2,\dots ,Q_n) \le {{\,\mathrm{MV}\,}}(Q_1,Q_2,\dots ,Q_n)$. The same applies for all arguments. Since $Q_i \subseteq \tilde{Q} := {{\,\mathrm{conv}\,}}(Q_1 \cup \dots \cup Q_n)$ for each $i=1,\dots ,n$, the inequality

$$\begin{aligned} {{\,\mathrm{MV}\,}}(Q_1,\dots ,Q_n) \le {{\,\mathrm{MV}\,}}(\tilde{Q},\dots ,\tilde{Q}) = n! {{\,\mathrm{vol}\,}}(\tilde{Q}) \end{aligned}$$

always holds regardless of the relative position of the polytopes. The present contribution shows that the equality can hold even when each $Q_i$ is strictly contained in $\tilde{Q}$.

B: Modifications to Polyhedral Homotopies

The apparent limitations of the construction of the polyhedral homotopy (22) are that the target system $P(\mathbf {x})$ is assumed to be in general position, zeros in $\mathbb {C}^n {\setminus } (\mathbb {C}^*)^n$ may not be reached, and the numerical condition of the equation $H(\mathbf {x},t) = \varvec{0}$ may be poor. These limitations are surmounted by modifications proposed in subsequent studies [44, 48, 54, 59, 74]. A commonly used extension of (22) with respect to the same liftings and target system is given by

where $c_{i,\mathbf {a}}$ and $\varepsilon _i$ are generic complex numbers and $B \mathbf {x}= (b_1 x_1,\dots ,b_n x_n)$ with $b_i \in \mathbb {R}^+$ is chosen to properly improve the numerical stability. It can be shown that as t varies from $-\infty$ to 0, the solutions of $H(\mathbf {x},t) = \varvec{0}$ also vary continuously forming smooth solution paths that collectively reach all isolated zeros of the target system $P(\mathbf {x})$ in $\mathbb {C}^n$. This extension has been adopted in PHoM [38], Hom4PS-2.0 [54], and Hom4PS-3 [21]. A variation of it can also be found in recent versions of PHCpack [77].

C: Libtropicana

The software package libtropicana⁶ is developed by the author specifically to carry out the experiments shown in Sect. 7. Given a convex polytope in $\mathbb {Z}^n$, it computes a regular subdivision and also produces the normalized volume of the polytope as a byproduct. It is based on a pivoting algorithm similar to the core algorithm of lrs [4]. But unlike lrs, which puts a special emphasis on memory efficiency and accuracy, libtropicana focuses on speed (potentially at the expense of higher memory consumption) and moderate sized polytopes. It is written completely in C++ with optional interface for leveraging BLAS and spBLAS (Sparse BLAS) routines. libtropicana is open source software. Users may freely distribute its source under the terms of the LGPL license.