King Chasing Problem in Chinese Chess is NP-hard

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.

Figures (15)

• Figure 1: The Chinese Chess board.
• Figure 2: The movement rules, intersections indicated by green arrows represent reachable intersections.
• Figure 3: The mate-in-one gadget.
• Figure 4: The start gadget.
• Figure 5: Directional cannons alternately check Black.

Theorems & Definitions (12)

• Definition 1: The king chasing problem in Chinese Chess
• Theorem 1
• Definition 2: 3-SAT problem
• Theorem 2: Introduction-to-the-Theory-of-Computation
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4