Planar 3-way Edge Perfect Matching Leads to A Holant Dichotomy

We prove a complexity dichotomy theorem for a class of Holant problems on planar 3-regular bipartite graphs. The complexity dichotomy states that for every weighted constraint function $f$ defining the problem (the weights can even be negative), the problem is either computable in polynomial time if $f$ satisfies a tractability criterion, or \#P-hard otherwise. One particular problem in this problem space is a long-standing open problem of Moore and Robson on counting Cubic Planar X3C. The dichotomy resolves this problem by showing that it is \numP-hard. Our proof relies on the machinery of signature theory developed in the study of Holant problems. An essential ingredient in our proof of the main dichotomy theorem is a pure graph-theoretic result: Excepting some trivial cases, every 3-regular plane graph has a planar 3-way edge perfect matching. The proof technique of this graph-theoretic result is a combination of algebraic and combinatorial methods. The P-time tractability criterion of the dichotomy is explicit. Other than the known classes of tractable constraint functions (degenerate, affine, product type, matchgates-transformable) we also identify a new infinite set of P-time computable planar Holant problems; however, its tractability is not by a direct holographic transformation to matchgates, but by a combination of this method and a global argument. The complexity dichotomy states that everything else in this Holant class is \#P-hard.