Complexity of a Tetris variant

TL;DR

This paper proves that maximizing cleared lines in a specific Tetris variant is NP-hard by reducing the 3-partition problem, addressing an open complexity question about the game.

Contribution

It provides the first complexity result for a variant of offline Tetris introduced by Demaine et al., showing NP-hardness through a reduction from 3-partition.

Findings

Maximizing cleared lines in this Tetris variant is NP-hard.

The problem remains hard even with unlimited moves in the first row.

This result answers an open question about Tetris complexity.

Abstract

In this paper we are going to solve an open problem about the game tetris. We are going to give the first results in the complexity of a variant of offline tetris introduced by Erik Demaine, Susan Hohenberger and David Liben Nowell in their paper "Tetris is hard, even to approximate". In this variant, that follows a model of movements introduced by John Brzustowsky, we can move and rotate a piece the number of times we want in the first row. But then, when we left the piece fall, we cannot move it or rotate it anymore. We are going to demonstrate that the problem of maximizing the number of cleared lines of this variant on a particular game board, is NP-hard by reducing the 3-partition problem to the problem of clearing the board in this variant of tetris