TETRAVEX is a widely played one person computer game in which you are given n2 unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a n by n square such that all neighbouring edges are labelled with an identical number. Unfortunately, playing TETRAVEX is computationally hard. More precisely, we prove that deciding if there is a tiling of the TETRA...

We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to...

In a Hiroimono puzzle, one must collect a set of stones from a square grid, moving along grid lines, picking up stones as one encounters them, and changing direction only when one picks up a stone. We show that deciding the solvability of such puzzles is NP-complete.

We prove that completing an untimed, unbounded track in TrackMania Nations Forever is NP-complete by using a reduction from 3-SAT and showing that a solution can be checked in polynomial time.

We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al., complete sets for all of the following complexity classes are m-mitotic: NP, coNP, ⊕P, PSPACE, and NEXP, as well as all levels of PH, MODPH, and the Boolean hierarchy over NP. In the c...

In a Hashiwokakero puzzle, one must build bridges to connect a set of islands. We show that deciding the solvability of such puzzles is NP-complete.

We study the computational complexity of a packing puzzle Fillmat, which is a type of pencil-and-paper puzzles made by Japanese puzzle publisher Nikoli. We show that the problem to decide if a given instance of Fillmat has a solution is NP-complete by a reduction from the circuit-satisfiability problem (Circuit-SAT). Our reduction is carefully designed so that we can also prove ASP-completeness...

For all k≥1, we show that deciding whether a graph is k-planar NP-complete, extending the well-known fact 1-planarity NP-complete. Furthermore, gap version of this decision problem In particular, given with local crossing number either at most k≥1 or least 2k, it NP-complete to decide k 2k. This algorithmic lower bound proves non-existence (2−ϵ)-approximation algorithm for any fixed k≥1. additi...