Super Mario Bros Proved NP-Hard

Back in January we looked at the work of Giovanni Viglietta at the University if Pisa in Italy who had proved the computational complexity of many computer games from the 80s and 90s, such as Pac-Man and Tron.

Viglietta pointed out that this kind of analysis is unnecessary for modern games because most incorporate Turing-equivalent scripting languages that easily allow the design of undecidable puzzles as part of the gameplay.

However, that still leaves many classics that are are as yet unclassifed. 

Today, Greg Aloupis at the Free University of Brussels in Belgium and a couple of buddies fill that hole, at least in part. 

These guys prove that several classic Nintendo games from the 80s are all NP-hard. The list includes the first three incarnations of Super Mario Bros, Donkey Kong and all the Legend of Zelda games. 

All these games are essentially the same in that they start at a specific point with the aim of reaching some goal. The question Aloupis and co ask is this: given the starting position, is it possible reach the goal? 

“If it is hard to decide even this question, then it is certainly hard to find an optimal path,” they say.

In this context, they go on to show that all the games are essentially versions of another problem called 3-SAT, which is known to be NP-complete. The process here is to show that 3SAT reduces to these problems in certain circumstances, thereby proving they are NP-hard.  

So if your youth was misspent playing Super Mario Bros, Donkey Kong or any of the other games these guys prove NP-Hard, then knowing how hard they actually were might provide a little comfort that you didn’t waste your time entirely. Then again, probably not!

Ref: arxiv.org/abs/1203.1895: Classic Nintendo Games are (NP-)Hard