Strings-and-Coins and Nimstring are PSPACE-complete

TL;DR

This paper proves that the combinatorial game Strings-and-Coins, which generalizes Dots-and-Boxes, is strongly PSPACE-complete on multigraphs, improving previous NP-hardness results and also applying to Nimstring variants.

Contribution

It establishes the PSPACE-completeness of Strings-and-Coins and Nimstring, providing a stronger complexity classification than prior NP-hardness results.

Findings

Strings-and-Coins is strongly PSPACE-complete.

The PSPACE-completeness applies to multigraphs.

The result extends to Nimstring variants.

Abstract

We prove that Strings-and-Coins -- the combinatorial two-player game generalizing the dual of Dots-and-Boxes -- is strongly PSPACE-complete on multigraphs. This result improves the best previous result, NP-hardness, argued in Winning Ways. Our result also applies to the Nimstring variant, where the winner is determined by normal play; indeed, one step in our reduction is the standard reduction (also from Winning Ways) from Nimstring to Strings-and-Coins.