A Tractability Gap Beyond Nim-Sums: It's Hard to Tell Whether a Bunch of Superstars Are Losers
arxiv(2024)
摘要
In this paper, we address a natural question at the intersection of
combinatorial game theory and computational complexity: "Can a sum of simple
tepid games in canonical form be intractable?" To resolve this fundamental
question, we consider superstars, positions first introduced in Winning Ways
where all options are nimbers. Extending Morris' classic result with hot games
to tepid games, we prove that disjunctive sums of superstars are intractable to
solve. This is striking as sums of nimbers can be computed in linear time. Our
analyses also lead to a family of elegant board games with intriguing
complexity, for which we present web-playable versions of the rulesets
described within.
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