In this paper, we study a combination (called the generalized cyclic Nimhoff) of the cyclic Nimhoff and subtraction games. We give the G-value of the game when all the G-value sequence of subtraction games have a common h-stair structure. 1

Universal partizan ruleset is a ruleset in which every game value of partizan games can be appear as a position. So far, generalized konane and turning tiles have been proved to be universal partizan rulesets. In this paper, we introduce two rulesets go on lattice and beyond the door and prove that they are universal partizan rulesets by using game tree preserving reduction.

In Combinatorial Game Theory, we study the set of games G, whose elements are mapped from positions of rulesets. In many case, given a ruleset, not all elements of G can be given as a position in the ruleset. It is an intriguing question what kind of ruleset would allow all of them to appear. In this paper, we introduce a ruleset named turning tiles and prove the ruleset is a universal partizan ruleset, that is, every element in G can occur as a position in the ruleset. This is the second universal partizan ruleset after generalized konane.

Large-scale quantum information processing requires the use of quantum error correcting codes to mitigate the effects of noise in quantum devices. Topological error-correcting codes, such as surface codes, are promising candidates as they can be implemented using only local interactions in a twodimensional array of physical qubits. Procedures such as defect braiding and lattice surgery can then be used to realize a fault-tolerant universal set of gates on the logical space of such topological codes. However, error correction also introduces a significant overhead in computation time, the number of physical qubits, and the number of physical gates. While optimizing fault-tolerant circuits to minimize this overhead is critical, the computational complexity of such optimization problems remains unknown. This ambiguity leaves room for doubt surrounding the most effective methods for compiling fault-tolerant circuits for a large-scale quantum computer. In this paper, we show that the optimization of a special subset of braided quantum circuits is NP-hard by a polynomial-time reduction of the optimization problem into a specific problem called PlanarRectilinear3SAT.

Given a set P of n points on which facilities can be placed and an integer k, we want to place k facilities on some points so that the minimum distance between facilities is maximized. The problem is called the k-dispersion problem. In this paper, we consider the 3-dispersion problem when P is a set of points on a plane (2-dimensional space). Note that the 2-dispersion problem corresponds to the diameter problem. We give an O(n) time algorithm to solve the 3-dispersion problem in the L ∞ metric, and an O(n) time algorithm to solve the 3-dispersion problem in the L 1 metric. Also, we give an O(n 2 log n) time algorithm to solve the 3-dispersion problem in the L 2 metric.