Finding a Hamiltonian Path in a Cube with Specified Turns is Hard

We prove the NP-completeness of finding a Hamiltonian path in an N × N × N cube graph with turns exactly at specified lengths along the path. This result establishes NP-completeness of Snake Cube puzzles: folding a chain of N3 unit cubes, joined at face centers (usually by a cord passing through all the cubes), into an N × N × N cube. Along the way, we prove a universality result that zig-zag chains (which must turn every unit) can fold into any polycube after 4 × 4 × 4 refinement, or into any Hamiltonian polycube after 2 × 2 × 2 refinement

The Traveling Salesman Problem Under Squared Euclidean Distances

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change

Four-dimensional understanding of quantum mechanics and Bell violation

While our natural intuition suggests us that we live in 3D space evolving in time, modern physics presents fundamentally different picture: 4D spacetime, Einstein's block universe, in which we travel in thermodynamically emphasized direction: arrow of time. Suggestions for such nonintuitive and nonlocal living in kind of "4D jello" come among others from: Lagrangian mechanics we use from QFT to GR saying that history between fixed past and future situation is the one optimizing action, special relativity saying that different velocity observers have different present 3D hypersurface and time direction, general relativity deforming shape of the entire spacetime up to switching time and space below the black hole event horizon, or the CPT theorem concluding fundamental symmetry between past and future. Accepting this nonintuitive living in 4D spacetime: with present moment being in equilibrium between past and future - minimizing tension as action of Lagrangian, leads to crucial surprising differences from intuitive "evolving 3D" picture, in which we for example conclude satisfaction of Bell inequalities - violated by the real physics. Specifically, particle in spacetime becomes own trajectory: 1D submanifold of 4D, making that statistical physics should consider ensembles like Boltzmann distribution among entire paths, what leads to quantum behavior as we know from Feynman's Euclidean path integrals or similar Maximal Entropy Random Walk (MERW). It results for example in Anderson localization, or the Born rule with squares - allowing for violation of Bell inequalities. Specifically, quantum amplitude turns out to describe probability at the end of half-spacetime from a given moment toward past or future, to randomly get some value of measurement we need to "draw it" from both time directions, getting the squares of Born rules.Comment: 13 pages, 18 figure

On the diffeomorphism commutators of lattice quantum gravity

We show that the algebra of discretized spatial diffeomorphism constraints in Hamiltonian lattice quantum gravity closes without anomalies in the limit of small lattice spacing. The result holds for arbitrary factor-ordering and for a variety of different discretizations of the continuum constraints, and thus generalizes an earlier calculation by Renteln.Comment: 16 pages, Te