A hard $n$-fold integral

Question

Consider the $n$-fold integral $$ J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} d\theta_n\ldots d\theta_2 d\theta_1 $$

whose intervals are defined by $$ \begin{align} I_1 = [0,1] \\ I_i = [\max(c_i,\theta_{i-1}),1] , 2\leq i\leq n \end{align} $$

and the $c_i \in [0,1]$ are predefined rational constants. Given a rational $v\in [0,1]$,

is deciding if $J=v$ NP-hard?

Informally , each $\max$ in the lower limits of the intervals leads to a two-way split in evaluating the integral, and thus to $2^{n-1}$ integrals that sum to $J$.

Answer 1

Your problem is a special case of the following: given a convex polyhedron defined by a set of linear inequalities, compute its volume. I believe that problem has been studied in the literature.

For instance, for related work on computing the volume of convex polyhedra/polytopes, see the following:

• Algorithm for finding the volume of a convex polytope

• Computing the Volume of a Convex Polytope Using Interval Arithmetic

• Polytope volume computation

There is probably lots more work out there that might be relevant, but maybe this gives you a start. You might start by reading that work to see if there are any techniques you can adapt; and by doing a literature review with the above as an entry point into the literature.

Answer 2

This question received an answer on MO ; the integral can be computed in $O(n^2)$ time.