Spider Solitaire is NP-Complete

TL;DR

This paper proves that a generalized version of Spider Solitaire is NP-Complete by reducing 3-SAT to the game, demonstrating its computational complexity and difficulty in solving optimally.

Contribution

It establishes the NP-Completeness of generalized Spider Solitaire, extending complexity results to this popular card game.

Findings

Spider Solitaire is NP-Complete in the generalized case

The proof reduces 3-SAT to Spider Solitaire

No polynomial-time algorithm exists for solving generalized Spider Solitaire unless P=NP

Abstract

This project investigates the potential of computers to solve complex tasks such as games. The paper proves that the complexity of a generalized version of spider solitaire is NP-Complete and uses much of structure of the proof that FreeCell is NP-Hard in the paper Helmert, M. "Complexity Results for Standard Benchmark Domains in Planning." Artificial Intelligence 143.2 (2003): 219-62. Print. A given decision problem falls in to the class NP-Complete if it is proven to be both in NP and in NP-Hard. To prove that this is the case the paper shows that, not only do the kinds of possible moves that can be reversed prove this, but it is also shown that no spider solitaire game of size n will take more than a polynomial number of moves to complete if such a completion is possible. The paper reduces 3-SAT to SpiderSolitaire (the name used throughout the proof when referring to the generalized version of popular solitaire variant "Spider Solitaire") by showing that any 3-SAT instance can be replicated using an appropriately arranged initial tableau. The example provided reinforces the proof of NP-Hardness and helps to make the proof easier to understand, but the definitive proof lies in the equations providing instruction on how to set up any 3-SAT instance of clause size C as a instance of SpiderSolitaire.