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An NP-hard generalization of Nim

Chunlei Liu

TL;DR

The paper introduces occupation games $(X,S,O)$ as a generalization of Nim and Subtraction and defines the Nash-equilibrium indicator ${\rm Truth}(A)\in\{0,1\}$. It analyzes the complexity of computing ${\rm Truth}(A)$, establishing NP-hardness via a subset-sum based reduction. The core contribution is a constructive reduction from Subset Sum that encodes the existence of a subset summing to $t$ into the truth value ${\rm Truth}(X)$. This work delineates the computational limits of equilibrium computation in broad classes of impartial games and underscores the intractability of generalizations of Nim.

Abstract

A new combinatorial game is given. It generalizes both Substraction and Nim. It is proved the computation of Nash equilibrium points in this new game is NP-hard.

An NP-hard generalization of Nim

TL;DR

The paper introduces occupation games as a generalization of Nim and Subtraction and defines the Nash-equilibrium indicator . It analyzes the complexity of computing , establishing NP-hardness via a subset-sum based reduction. The core contribution is a constructive reduction from Subset Sum that encodes the existence of a subset summing to into the truth value . This work delineates the computational limits of equilibrium computation in broad classes of impartial games and underscores the intractability of generalizations of Nim.

Abstract

A new combinatorial game is given. It generalizes both Substraction and Nim. It is proved the computation of Nash equilibrium points in this new game is NP-hard.
Paper Structure (2 sections, 5 theorems, 49 equations)

This paper contains 2 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. ${\rm NP}$-hardness

Key Result

Theorem 1.7

Let $(X,S,O)$ be an occupation game and $A\in S$. Then the computation of ${\rm Truth}(A)$ is ${\rm NP}$-hard.

Theorems & Definitions (12)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4: Bou02,Gru39
  • Example 1.5
  • Example 1.6: Bou02
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 2 more