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Published online by Cambridge University Press: 30 May 2025
A pair of integer sequences that split ℤ>0 is often—especially in the context of combinatorial game theory—defined recursively by
an = mex {ai , ai: 0 ≤ i ≤ n},bn = an+cn (n≥ 0),
where mex (Minimum EXcludant) of a subset S of nonnegative integers is the smallest nonnegative integer not in S, and c : ℤ≥0→ℤ0. Given x, y ∈ ℤ≥0, a typical problem is to decide whether x = an, y = bn. For general functions c, the best algorithm for this decision problem was until now exponential in the input size Ω(log x +log y). We prove constructively that the problem is actually polynomial for the wide class of approximately linear functions cn. This solves constructively and efficiently the complexity question of a number of previously analyzed take-away games of various authors.
This paper is about the complexity of combinatorial games. Its main contribution is showing constructively that a large class of games whose complexity was hitherto unknown and its best winning strategy was exponential, is actually solvable in polynomial time.
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