On Planar Valued CSPs

TL;DR

This paper investigates the complexity of planar valued constraint satisfaction problems (VCSPs), revealing conditions for tractability and providing a complete classification for conservative cases, extending previous results from CSPs.

Contribution

It extends known results from CSPs to VCSPs, showing intractable Boolean VCSPs must be self-complementary for tractability in planar cases, and offers a full classification of conservative planar VCSPs.

Findings

Intractable Boolean VCSPs must be self-complementary to be tractable in planar settings.

Planarity does not introduce new tractable cases for conservative VCSPs.

Complete classification of conservative planar VCSPs on finite domains.

Abstract

We study the computational complexity of planar valued constraint satisfaction problems (VCSPs), which require the incidence graph of the instance be planar. First, we show that intractable Boolean VCSPs have to be self-complementary to be tractable in the planar setting, thus extending a corresponding result of Dvorak and Kupec [ICALP'15] from CSPs to VCSPs. Second, we give a complete complexity classification of conservative planar VCSPs on arbitrary finite domains. In this case planarity does not lead to any new tractable cases and thus our classification is a sharpening of the classification of conservative VCSPs by Kolmogorov and Zivny [JACM'13].