  Let $\mathcal{K}$ be a finite pure simplicial $d$-complex, with oriented
facets $\{F_i\}$, which is boundaryless in the sense that $\sum\partial F_i=0$.
We call such a $\mathcal{K}$ an \textit{admissible $d$-complex}. Given an
admissible $d$-complex, one can ask for the smallest collection $\{T_i\}$ of
oriented $(d+1)$-simplices on the vertices of $\mathcal{K}$ which decomposes
$\mathcal{K}$ in the sense that $\sum \partial T_i = \mathcal{K}$. Let the
minimum size of such a collection be $V_\mathbb{Z}(\mathcal{K})$, and let
$V_\mathbb{Q}(\mathcal{K})$ be the relaxed analog where fractional
$(d+1)$-simplices may be used. We explain how these quantities may be computed
via integer and linear programming, and show how lower bounds may be obtained
by exploiting LP-duality. We then prove that $V_\mathbb{Q}$ and $V_\mathbb{Z}$
are both additive under disjoint union and connected sum along a $d$-simplex.
The remainder of the paper explores integrality gaps between $V_\mathbb{Z}$ and
$V_\mathbb{Q}$ in dimension 1, where we share what we believe is the simplest
admissible complex with an integrality gap; and in dimension 2, where we
collect some results on integrality gaps for triangulations of the 2-sphere for
a companion paper with Zili Wang and Peter Doyle.
