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A Tractability Gap Beyond Nim-Sums: It’s Hard to Tell Whether a Bunch of Superstars Are Losers

Authors Kyle Burke , Matthew Ferland , Svenja Huntemann , Shanghua Teng

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Author Details

Kyle Burke
  • Florida Southern College, Lakeland, FL, USA
Matthew Ferland
  • University of Southern California, Los Angeles, CA, USA
Svenja Huntemann
  • Mount Saint Vincent University, Halifax, Canada
Shanghua Teng
  • University of Southern California, Los Angeles, CA, USA

Funding

  • Teng, Shanghua: Supported in part by NSF Grant CCF-2308744 and the Simons Investigator Award from the Simons Foundation.

Cite As Get BibTex

Kyle Burke, Matthew Ferland, Svenja Huntemann, and Shanghua Teng. A Tractability Gap Beyond Nim-Sums: It’s Hard to Tell Whether a Bunch of Superstars Are Losers. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FUN.2024.8

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Combinatorial Game Theory
  • NP-hardness
  • Superstars

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