King Chasing Problem in Chinese Chess is NP-hard

2026-04-02

AI Summary

This study investigates the computational complexity of the "generalized king chase" problem in Chinese chess, which asks whether the first player can always force a checkmate by delivering consecutive checks on an $n \times n$ board. For the first time, we rigorously establish that this problem is NP-hard by presenting a polynomial-time reduction from 3-SAT. Our result uncovers the intrinsic computational difficulty inherent in the king chase scenario, thereby filling a notable gap in the theory of combinatorial game complexity for Chinese chess variants. This theoretical advance provides a foundational basis for the design of related game rules and the analysis of associated algorithms.

Abstract

We prove that king chasing problem in Chinese Chess is NP-hard when generalized to $n\times n$ boards. `King chasing' is a frequently-used strategy in Chinese Chess, which means that the player has to continuously check the opponent in every move until finally checkmating the opponent's king. The problem is to determine which player has a winning strategy in generalized Chinese Chess, under the constraints of king chasing. Obviously, it is a sub-problem of generalized Chinese Chess problem. We prove that king chasing problem in Chinese Chess is NP-hard by reducing from the classic NP-complete problem 3-SAT.

Problem

• Chinese Chess
• king chasing
• NP-hard
• generalized game
• winning strategy

Innovation

• NP-hard
• Chinese Chess
• king chasing
• 3-SAT reduction
• computational complexity