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

TL;DR

It is proved that disjunctive sums of superstars are intractable to solve, a natural question at the intersection of combinatorial game theory and computational complexity.

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.

Figures (4)

• Figure 1: A Paint Can position consisting of two stacks of bricks with value $\left\{\ 0, \ast 2, \ast 4\ \middle|\ \ast, \ast 2\ \right\}$ + $\ast 4$. In the leftmost stack, the Blue player may choose to remove either of the blue bricks or the green brick. The Red player may choose to remove either the red or green brick. Neither player may choose to remove the gray brick. In the rightmost stack, all bricks are already green, so no can of paint is necessary. If the Blue player removes the top brick from the left stack, the result will be a stack of four green bricks, as is in the right stack.
• Figure 2: A Paint Can position equal to $\left\{\ 0, \ast 1\ \middle|\ 0, \ast 2\ \right\} + \left\{\ \ast 2\ \middle|\ \ast 3\ \right\} + \left\{\ 0, \ast, \ast 2\ \middle|\ \ast\ \right\}$.
• Figure 3: Multi-state variable $x_a$ with four possible states: $s_1$, $s_2$, $s_3$, $s_4$. The overall color indicates that the chosen state is $s_2$.
• Figure 4: An Example Game of Blackout

Theorems & Definitions (18)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Proposition 2.3: ONAG:2000
• Definition 2.4: Paint Can
• Definition 3.1
• Corollary 3.2: No-0 games
• Lemma 3.3: 0 game win
• proof
• Theorem 3.4