Computational Complexity of Swish

2026-01-14

AI Summary

This study investigates the computational complexity of the original Swish card game—where each card contains two symbols—under various geometric transformations. Focusing on the decision problem of whether a valid swish (i.e., a legal overlay of cards) exists, we employ techniques from computational complexity theory and reductions, along with precise modeling of transformations such as flipping and rotation. We establish for the first time that the problem becomes NP-complete when either single-axis flipping or 180-degree rotation is permitted, whereas it admits a polynomial-time algorithm in the absence of any geometric transformations. Our work fully characterizes the complexity landscape of the two-symbol Swish problem under transformation constraints, providing an exact classification based on the interplay between symbol count and allowed operations.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Swish is a card game in which players are given cards having symbols (hoops and balls), and find a valid superposition of cards, called a"swish."Dailly, Lafourcade, and Marcadet (FUN 2024) studied a generalized version of Swish and showed that the problem is solvable in polynomial time with one symbol per card, while it is NP-complete with three or more symbols per card. In this paper, we resolve the previously open case of two symbols per card, which corresponds to the original game. We show that Swish is NP-complete for this case. Specifically, we prove the NP-hardness when the allowed transformations of cards are restricted to a single (horizontal or vertical) flip or 180-degree rotation, and extend the results to the original setting allowing all three transformations. In contrast, when neither transformation is allowed, we present a polynomial-time algorithm. Combining known and our results, we establish a complete characterization of the computational complexity of Swish with respect to both the number of symbols per card and the allowed transformations.

Problem

• Swish
• computational complexity
• NP-completeness
• card game
• symbol superposition

Innovation

• computational complexity
• NP-completeness
• Swish game
• polynomial-time algorithm
• card transformations