Threes!, Fives, 1024!, and 2048 are Hard

We analyze the computational complexity of the popular computer games Threes!, 1024!, 2048 and many of their variants. For most known versions expanded to an m x n board, we show that it is NP-hard to decide whether a given starting position can be played to reach a specific (constant) tile value.Comment: 14 pages, 9 figure

Vortex-strings in N=2 quiver X U(1) theories

We study half-BPS vortex-strings in four dimensional N=2 supersymmetric quiver theories with gauge group SU(N)^n X U(1). The matter content of the quiver can be represented by what we call a tetris diagram, which simplifies the analysis of the Higgs vacua and the corresponding strings. We classify the vacua of these theories in the presence of a Fayet-Iliopoulos term, and study strings above fully-Higgsed vacua. The strings are studied using classical zero modes analysis, supersymmetric localization and, in some cases, also S-duality. We analyze the conditions for bulk-string decoupling at low energies. When the conditions are satisfied, the low energy theory living on the string's worldsheet is some 2d N=(2,2) supersymmetric non-linear sigma model. We analyze the conditions for weak to weak 2d-4d map of parameters, and identify the worldsheet theory in all the cases where the map is weak to weak. For some SU(2) quivers, S-duality can be used to map weakly coupled worldsheet theories to strongly coupled ones. In these cases, we are able to identify the worldsheet theories also when the 2d-4d map of parameters is weak to strong.Comment: 61 pages, 10 figure

Approximate Dynamic Programming via a Smoothed Linear Program

We present a novel linear program for the approximation of the dynamic programming cost-to-go function in high-dimensional stochastic control problems. LP approaches to approximate DP have typically relied on a natural “projection” of a well-studied linear program for exact dynamic programming. Such programs restrict attention to approximations that are lower bounds to the optimal cost-to-go function. Our program—the “smoothed approximate linear program”—is distinct from such approaches and relaxes the restriction to lower bounding approximations in an appropriate fashion while remaining computationally tractable. Doing so appears to have several advantages: First, we demonstrate bounds on the quality of approximation to the optimal cost-to-go function afforded by our approach. These bounds are, in general, no worse than those available for extant LP approaches and for specific problem instances can be shown to be arbitrarily stronger. Second, experiments with our approach on a pair of challenging problems (the game of Tetris and a queueing network control problem) show that the approach outperforms the existing LP approach (which has previously been shown to be competitive with several ADP algorithms) by a substantial margin