On a dispersion problem in grid labeling

International audienceGiven $k$ labelings of a finite $d$-dimensional cubical grid, define the combined distance between two labels to be the sum of the $l_1$-distance between the two labels in each labeling. We want to construct $k$ labelings which maximize the minimum combined distance between any two labels. When $d = 1$, this can be interpreted as placing $n$ nonattacking rooks in a $k$-dimensional chessboard of size $n$ in such a way to maximize the minimum $l_1$-distance between any two rooks. Rook placements are also known as Latin hypercube designs in the literature. In this paper, we revisit this problem with a more geometric approach. Instead of providing explicit but complicated formulas, we construct rook placements in a $k$-dimensional chessboard of size $n$ as certain lattice-like structures for certain well-chosen values of $n$. Then, we extend these constructions to any values of $n$ using geometric arguments. With this method, we present a clean and geometric description of the known optimal rook placements in the two-dimensional square grid. Furthermore, we provide asymptotically optimal constructions of $k$ labelings of $d$-dimensional cubical grids which maximize the minimum combined distance. Finally, we discuss the extension of this problem to labelings of an arbitrary graph. We prove that deciding whether a graph has two labelings with combined distance at least 3 is at least as hard as graph isomorphism

P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside $[0,1]^n$, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in $\{0,1\}^n$, we get geometric formulations as the question if plane or sphere intersects with $\{0,1\}^n$. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of $\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi$ fourier-type integral for some natural $k_i$. The last discussed approach is using anti-commuting Grassmann numbers $\theta_i$, making $(A \cdot \textrm{diag}(\theta_i))^n$ nonzero only if $A$ has a Hamilton cycle. Hence, the P$\ne$NP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure