Kernelizing buttons and scissors

Buttons & Scissors (B&S) is a popular logic game, where we are given an n-by-m grid with some cells being sewed with colored buttons, and the objective is to decide whether we can empty the grid by certain scissor cuts. Recently, the computational complexity of this problem has attracted a lot of attention, and the problem, together with several of its restrictions, has been shown to be NP-complete. In this paper we study this problem in the realm of parameterized complexity. In particular, we show that B&S, when restricted to using only horizontal and vertical scissor cuts, is FPT parameterized by the number of cuts required to empty the grid (say k). Precisely, we design an algorithm that runs in time 2^O(k log k⁾ + (n + m)^O(1) and decides whether there is a wining sequence with at most k scissor cuts. At the heart of our algorithm is a polynomial time kernelization procedure, that, given an instance of the game on a n-by-m board, obtains an equivalent instance on a board of dimension O(k²)-by-O(k²) that has at most O(k³) buttons. When the input is restricted to n-by-1 grid, we get a kernel with board size O(k)-by-1 and an algorithm with running time 2^O(k⁾ + n^O(1). Finally, in the general setting, that is when diagonal scissor cuts are also allowed, we show that the problem is FPT parameterized by k and r; where r is an upper bound on the length of any diagonal cut.