4 research outputs found
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.
Holographic Algorithm with Matchgates Is Universal for Planar CSP Over Boolean Domain
We prove a complexity classification theorem that classifies all counting
constraint satisfaction problems (CSP) over Boolean variables into exactly
three categories: (1) Polynomial-time tractable; (2) P-hard for general
instances, but solvable in polynomial-time over planar graphs; and (3)
P-hard over planar graphs. The classification applies to all sets of local,
not necessarily symmetric, constraint functions on Boolean variables that take
complex values. It is shown that Valiant's holographic algorithm with
matchgates is a universal strategy for all problems in category (2).Comment: 94 page