The Complexity of Planar Boolean #CSP with Complex Weights

TL;DR

This paper establishes a complexity dichotomy for symmetric complex-weighted Boolean #CSP problems on planar graphs, identifying which are tractable via holographic reductions and which are #P-hard, extending prior real-weight results.

Contribution

It generalizes the #CSP dichotomy to complex weights on planar graphs and introduces new proof techniques for analyzing complexity in this setting.

Findings

Identifies #P-hard and tractable classes of planar symmetric complex-weighted #CSP.

Extends the planar #CSP dichotomy theorem from real to complex weights.

Reduces evaluating the Tutte polynomial at (3,3) to counting Eulerian orientations, proving #P-hardness.

Abstract

We prove a complexity dichotomy theorem for symmetric complex-weighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #P-hard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3,3) to counting the number of Eulerian orientations over planar 4-regular graphs to show the latter is #P-hard. This strengthens a theorem by Huang and Lu to the planar setting. Our proof techniques combine new ideas with refinements and extensions of existing techniques. These include planar pairings, the recursive unary construction, the anti-gadget technique, and pinning in the Hadamard basis.