Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$ we canonically associate permanent of $M$ to the number of perfect matchings in the graph.

Is there a bipartite graph which is balanced we can associate to the square of the determinant or absolute value of the determinant in a canonical manner without computing the determinant explicitly?

Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$ we canonically associate permanent of $M$ to the number of perfect matchings in the graph.

Is there a bipartite graph which is balanced we can associate to the square of the determinant or absolute value of the determinant in a canonical manner without computing the determinant explicitly?

Update Observe the construction of the graph is non-canonical in any known way. It has to be clever.

Integer Linear Program in $d$ integer variables, $m$ constraints and representable in $L$ bits is in $O(d^{\mathsf{poly}(d)}mL)$ time by an algorithm of Lenstra in https://pubsonline.informs.org/doi/abs/10.1287/moor.8.4.538. Hence in $d$ fixed dimensions it is in $\mathsf P$

Shortest vector problem in a lattice in $k$ dimensions is $\mathsf{NP}$-hard under randomized reductions.

Is there any reduction from $d$ integer variables, $m$ constraints and representable in $L$ bits integer linear programming to $k=O(\mathsf{poly}(dmL))$-dimensional lattice shortest vector problem?

Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$ we canonically associate permanent of $M$ to the number of perfect matchings in the graph.

Is there a bipartite graph which is balanced we can associate to the square of the determinant or absolute value of the determinant in a canonical manner without computing the determinant explicitly?