# A hard $n$-fold integral

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 $J$.

• I wonder if there are otherways to find the volume of this polytope. Have you consider math.stackExchange? Commented Jun 27, 2013 at 12:11
• I took a call on this question and posted a copy on MO: mathoverflow.net/questions/135076/an-np-hard-n-fold-integral
– PKG
Commented Jun 28, 2013 at 5:24

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:

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.

• Not to forget: The problem is #P complete as shown by Dyer and Frieze (1988). Commented Jun 28, 2013 at 8:15

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