Relevance-Zone Reduction in Game Solving

TL;DR

The paper addresses the challenge of solving games with exponentially large search trees by exploiting localized regions of relevance. It introduces an iterative Relevance-Zone Reduction (RZR) framework that constrains search regions and guides solvers toward smaller, more reusable RZs, supported by three constraint-generation strategies and an enhanced Relevance-Zone Pattern Table (RZT). Empirical results on 7×7 Killall-Go show the average RZ size shrinking to 85.95% of its original, with reduced RZs stored as reusable knowledge for future solving tasks, including larger boards or different openings. The approach advances practical game solving by enabling more efficient search, stronger pattern reuse, and transferability of learned constraints across tasks.

Abstract

Game solving aims to find the optimal strategies for all players and determine the theoretical outcome of a game. However, due to the exponential growth of game trees, many games remain unsolved, even though methods like AlphaZero have demonstrated super-human level in game playing. The Relevance-Zone (RZ) is a local strategy reuse technique that restricts the search to only the regions relevant to the outcome, significantly reducing the search space. However, RZs are not unique. Different solutions may result in RZs of varying sizes. Smaller RZs are generally more favorable, as they increase the chance of reuse and improve pruning efficiency. To this end, we propose an iterative RZ reduction method that repeatedly solves the same position while gradually restricting the region involved, guiding the solver toward smaller RZs. We design three constraint generation strategies and integrate an RZ Pattern Table to fully leverage past solutions. In experiments on 7x7 Killall-Go, our method reduces the average RZ size to 85.95% of the original. Furthermore, the reduced RZs can be permanently stored as reusable knowledge for future solving tasks, especially for larger board sizes or different openings.

Figures (5)

• Figure 1: Workflow of iterative RZ reduction (RZR). Top-row: constraint for each round—RZ from the last solved round (green), locked outer-region (red), newly banned point (yellow, overlap with RZ). Bottom-row: RZ returned by the solver under that constraint (unshaded if unsolved).
• Figure 2: Four openings from wu_game_2024.
• Figure 3: Early Stopping.
• Figure 4: Irreducible positions.
• Figure 5: Sample problem from the DA opening. Initial position (upper‑left); root RZs w/o and w/ RZR (middle‑left, bottom‑left). Columns 1–4: representative branches—$\times$ baseline white move, $\triangle$ RZR‑guided move (top row). Shaded cells mark the minimal leaf‑level RZs; their union produces the RZs in Column 1. Moves 25 and 31 are played at the same point in the third row, third column. For Case 4, both the baseline and RZR play at the same position ($\Box$).