FKT algorithm

Abstract: We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. In the study of counting complexity, such as #CSP, there are problems which are #P-hard over general graphs but P-time solvable over planar graphs. A recurring theme has been that a holographic reduction [Val08] to FKT precisely captures these problems. This dichotomy answers the question: Is this a universal strategy? Surprisingly, we discover new planar tractable problems in the Holant framework (which generalizes #CSP) that are not expressible by a holographic reduction to FKT. In particular, the putative form of a dichotomy for planar Holant problems is false. Nevertheless, we prove a dichotomy for #CSP2, a variant of #CSP where every variable appears even times, that the presumed universality holds for #CSP2. This becomes an important tool in the proof of the full dichotomy, which refutes this universality in general. The full dichotomy says that the new P-time algorithms and the strategy of holographic reductions to FKT together are universal for these locally defined counting problems. As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hyper graphs is P-time computable when the incidence graph is planar and k a#x2265; =5. The same problem is #P-hard when k=3 or k=4, also a consequence of the dichotomy. More generally, over hyper graphs with specified hyper edge sizes and the same planarity assumption, #PM is P-time computable if the greatest common divisor (gcd) of all hyper edge sizes is at least 5.

Abstract: We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for the undirected graphs. We prove that the task of computing the undirected determinants as well as permanents for planar graphs, whose vertices have degree at most 4, is \#P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains \#P-complete while the permanent can be reduced to the FKT algorithm, and therefore is polynomial. The undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. The concept of the undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time.