On the Learnability of Shuffle Ideals

PAC learning of unrestricted regular languages is long known to be a difficult problem. The class of shuffle ideals is a very restricted subclass of regular languages, where the shuffle ideal generated by a string u is the collection of all strings containing u as a subsequence. This fundamental language family is of theoretical interest in its own right and provides the building blocks for other important language families. Despite its apparent simplicity, the class of shuffle ideals appears quite difficult to learn. In particular, just as for unrestricted regular languages, the class is not properly PAC learnable in polynomial time if RP 6= NP, and PAC learning the class improperly in polynomial time would imply polynomial time algorithms for certain fundamental problems in cryptography. In the positive direction, we give an efficient algorithm for properly learning shuffle ideals in the statistical query (and therefore also PAC) model under the uniform distribution.T-Party Projec

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

Two Hands Are Better Than One (up to constant factors): Self-Assembly In The 2HAM vs. aTAM

We study the difference between the standard seeded model (aTAM) of tile self-assembly, and the "seedless" two-handed model of tile self-assembly (2HAM). Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model.National Science Foundation (U.S.) (NSF grant CDI-0941538

Folding equilateral plane graphs

22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. ProceedingsWe consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, such reconfiguration is known to be impossible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Not only is the equilateral constraint necessary for this result, but we show that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state with a specified “outside region”. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete

Total Tetris: Tetris with Monominoes, Dominoes, Trominoes, Pentominoes,...

We consider variations on the classic video game Tetris where pieces are k-ominoes instead of the usual tetrominoes (k = 4), as popularized by the video games ntris and Pentris. We prove that it is NP-complete to survive or clear a given initial board with a given sequence of pieces for each k ≤ 5, complementing the previous NP-completeness result for k = 4. More surprisingly, we show that board clearing is NP-complete for k = 3; and if pieces may not be rotated, then clearing is NP-complete for k = 2 and survival is NP-complete for k = 3. All of these problems can be solved in polynomial time for k = 1

Free Edge Lengths in Plane Graphs

We study the impact of metric constraints on the realizability of planar graphs. Let G be a subgraph of a planar graph H (where H is the “host” of G). The graph G is free in H if for every choice of positive lengths for the edges of G, the host H has a planar straight-line embedding that realizes these lengths; and G is extrinsically free in H if all constraints on the edge lengths of G depend on G only, irrespective of additional edges of the host H. We characterize the planar graphs G that are free in every host H, G⊆H, and all the planar graphs G that are extrinsically free in every host H, G⊆H. The case of cycles G=C[subscript k] provides a new version of the celebrated carpenter’s rule problem. Even though cycles C[subscript k], k≥4, are not extrinsically free in all triangulations, it turns out that “nondegenerate” edge lengths are always realizable, where the edge lengths are considered degenerate if the cycle can be flattened (into a line) in two different ways. Separating triangles, and separating cycles in general, play an important role in our arguments. We show that every star is free in a 4-connected triangulation (which has no separating triangle)