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Friends-and-strangers is PSPACE-complete

Chao Yang, Zhujun Zhang

TL;DR

We address the problem Fas for the configuration-reachability in the friends-and-strangers graph. We provide a $PSPACE$-membership proof and a planar reduction from $NCL$ using gadget constructions to simulate edge flips. We prove $PSPACE$-hardness via edge and vertex gadgets that encode edge directions, with eight colors to model friendship relations, yielding a planar $X$ with max degree $3$ and corresponding $Y$. Thus Fas is $PSPACE$-complete, highlighting the intractability of reconfiguration in the friends-and-strangers model under planar, bounded-degree constraints.

Abstract

In this paper, we show that the friends-and-strangers problem is PSPACE-complete by reduction from the Ncl (non-deterministic constraint logic) problem.

Friends-and-strangers is PSPACE-complete

TL;DR

We address the problem Fas for the configuration-reachability in the friends-and-strangers graph. We provide a -membership proof and a planar reduction from using gadget constructions to simulate edge flips. We prove -hardness via edge and vertex gadgets that encode edge directions, with eight colors to model friendship relations, yielding a planar with max degree and corresponding . Thus Fas is -complete, highlighting the intractability of reconfiguration in the friends-and-strangers model under planar, bounded-degree constraints.

Abstract

In this paper, we show that the friends-and-strangers problem is PSPACE-complete by reduction from the Ncl (non-deterministic constraint logic) problem.
Paper Structure (2 sections, 2 theorems, 1 equation, 6 figures)

This paper contains 2 sections, 2 theorems, 1 equation, 6 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm_main']}

Key Result

Theorem 1

The Fas problem is $\textsf{PSPACE}$-complete, even if the location graph $X$ is a planar graph with maximum degree $3$.

Figures (6)

  • Figure 1: Ncl (a) an AND vertex, (b) an OR vertex
  • Figure 2: Friends and strangers relations
  • Figure 3: Blue edge gadget (blue edge of NCL, with weight $2$)
  • Figure 4: Red edge gadget (red edge of NCL, with weight $1$)
  • Figure 5: The OR vertex gadget
  • ...and 1 more figures

Theorems & Definitions (5)

  • Definition 1: The Fas Problem
  • Theorem 1
  • Definition 2: hd05, The configuration-to-configuration Ncl Problem
  • Theorem 2: hd05
  • proof : Proof of Theorem \ref{['thm_main']}