2 research outputs found
A Holant Dichotomy: Is the FKT Algorithm Universal?
We prove a complexity dichotomy for complex-weighted Holant problems with an
arbitrary set of symmetric constraint functions on Boolean variables. This
dichotomy is specifically to answer the question: Is the FKT algorithm under a
holographic transformation a \emph{universal} strategy to obtain
polynomial-time algorithms for problems over planar graphs that are intractable
in general? This dichotomy is a culmination of previous ones, including those
for Spin Systems, Holant, and #CSP. A recurring theme has been that a
holographic reduction to FKT is a universal strategy. Surprisingly, for planar
Holant, we discover new planar tractable problems that are not expressible by a
holographic reduction to FKT.
In previous work, an important tool was a dichotomy for #CSP^d, which denotes
#CSP where every variable appears a multiple of d times. However its proof
violates planarity. We prove a dichotomy for planar #CSP^2. We apply this
planar #CSP^2 dichotomy in the proof of the planar Holant dichotomy.
As a special case of our new planar tractable problems, counting perfect
matchings (#PM) over k-uniform hypergraphs is polynomial-time computable when
the incidence graph is planar and k >= 5. The same problem is #P-hard when k=3
or k=4, which is also a consequence of our dichotomy. When k=2, it becomes #PM
over planar graphs and is tractable again. More generally, over hypergraphs
with specified hyperedge sizes and the same planarity assumption, #PM is
polynomial-time computable if the greatest common divisor of all hyperedge
sizes is at least 5.Comment: 128 pages, 36 figure
The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems
We show that an effective version of Siegel’s Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the κ ≥ 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. This dichotomy holds even when restricted to planar graphs. A special case of this result is that counting edge κ-colorings is #P-hard over planar 3-regular graphs for κ ≥ 3. In fact, we prove that counting edge κ-colorings is #P-hard over planar r-regular graphs for all κ ≥ r ≥ 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.