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Computational Complexity of Motion Planning of a Robot through Simple Gadgets

Any single nontrivial four-location two-state gadget type is enough for motion planning to become PSPACE-complete, while any set of simpler gadgets (effectively two-location or one-state) has a polynomial-time motion planning algorithm.
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A simple proof that the $(n^2-1)$-puzzle is hard

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Tree-Residue Vertex-Breaking: a new tool for proving hardness

A simple proof of the known result that Hamiltonicity in max-degree-$3$ square grid graphs is NP-hard is given, and a connection between TRVB and the Hypergraph Spanning Tree problem is demonstrated.
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18. Clickomania Is Hard, Even with Two Colors and Columns

Clickomania is a classic computer puzzle game (also known as SameGame, Chain-Shot!, and Swell-Foop, among other names). Originally developed by Kuniaki “Morisuke” Moribe under the name Chain-Shot!…
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Hamiltonicity is Hard in Thin or Polygonal Grid Graphs, but Easy in Thin Polygonal Grid Graphs

This paper considers a new restriction, where the grid graph is both thin and polygonal, and proves that Hamiltonicity then becomes polynomially solvable for square, triangular, and hexagonal grid graphs.
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Hardness of Token Swapping on Trees

A broad class of algorithms is identified that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is 2) and no such algorithm can achieve an approximation factor less Computational co-organized Erik 22–29, Holetown, Barbados.
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Solving the Rubik's Cube Optimally is NP-complete

In this paper, we prove that optimally solving an $n \times n \times n$ Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous…
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Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

It is shown that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies.
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Circumscribing Polygons and Polygonizations for Disjoint Line Segments

It is proved that every arrangement of disjoint line segments in the plane has a subset of size $\Omega(\sqrt{n})$ that admits a circumscribing polygon, which is the first improvement on this bound in 20 years.
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Computational complexity of numberless Shakashaka

This paper shows that Shakashaka is NP-complete in the case of numberless black squares.
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