193 research outputs found
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 defining the problem (the weights can
even be negative), the problem is either computable in polynomial time if
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.Comment: arXiv admin note: text overlap with arXiv:2110.0117
Hypohamiltonian and almost hypohamiltonian graphs
This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results
Classical Ising model test for quantum circuits
We exploit a recently constructed mapping between quantum circuits and graphs
in order to prove that circuits corresponding to certain planar graphs can be
efficiently simulated classically. The proof uses an expression for the Ising
model partition function in terms of quadratically signed weight enumerators
(QWGTs), which are polynomials that arise naturally in an expansion of quantum
circuits in terms of rotations involving Pauli matrices. We combine this
expression with a known efficient classical algorithm for the Ising partition
function of any planar graph in the absence of an external magnetic field, and
the Robertson-Seymour theorem from graph theory. We give as an example a set of
quantum circuits with a small number of non-nearest neighbor gates which admit
an efficient classical simulation.Comment: 17 pages, 2 figures. v2: main result strengthened by removing
oracular settin
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