New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification

We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models on planar graphs, assuming #P is not P. This leads to the following exact complexity classification: For every parameter setting in ${\mathbb C}$ for the six-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) #P-hard for general graphs but computable in P-time for planar graphs, or (3) #P-hard even for planar graphs. The classification has an explicit criterion. The new P-time cases in (2) provably cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local connection to #CSP, defined in terms of a "loop space". This is the first substantive advance toward a planar Holant classification with not necessarily symmetric constraints. We introduce M\"obius transformation on ${\mathbb C}$ as a powerful new tool in hardness proofs for counting problems.Comment: 61 pages, 16 figures. An extended abstract appears in SODA 202

On Two-Dimensional Mesh Networks and Their Simulation with P Systems

We analize in this paper the possibility of simulating the parallel architecture SIMD-MC2, also known as the two-dimensional mesh, with P systems with dynamic communication graphs. We illustrate this simulation for an algorithm which computes the sum of given integers. Next, we show how to extend the formalism to the reduction problem

Mark Sequences In Digraphs

In Chapter 1, we present a brief introduction of digraphs and some def- initions. Chapter 2 is a review of scores in tournaments and oriented graphs. Also we have obtained several new results on oriented graph scores and we have given a new proof of Avery's theorem on oriented graph scores. In chap- ter 3, we have introduced the concept of marks in multidigraphs, non-negative integers attached to the vertices of multidigraphs. We have obtained several necessary and su cient conditions for sequences of non-negative integers to be mark sequences of some r-digraphs. We have derived stronger inequalities for these marks. Further we have characterized uniquely mark sequences in r-digraphs. This concept of marks has been extended to bipartite multidi- graphs and multipartite multidigraphs in chapter 4. There we have obtained characterizations for mark sequences in these types of multidigraphs and we have given algorithms for constructing corresponding multidigraphs. Chap- ter 5 deals with imbalances and imbalance sequences in digraphs. We have generalized the concept of imbalances to oriented bipartite graphs and have obtained criteria for a pair of integers to be the pair of imbalance sequences of some oriented bipartite graph. We have shown the existence of an oriented bipartite graph whose imbalance set is the given set of integers

Structure and dynamics of Saturn's outer magnetosphere and boundary regions

In 1979-1981, the three USA spacecraft Pioneer 11 and Voyagers 1 and 2 discovered and explored the magnetosphere of Saturn to the limited extent possible on flyby trajectories. Considerable variation in the locations of the bow shock (BS) and magnetopause (MP) surfaces were observed in association with variable solar wind conditions and, during the Voyager 2 encounter, possible immersion in Jupiter's distant magnetic tail. The limited number of BS and MP crossings were concentrated near the subsolar region and the dawn terminator, and that fact, together with the temporal variability, makes it difficult to assess the three dimensional shape of the sunward magnetospheric boundary. The combined BS and MP crossing positions from the three spacecraft yield an average BS-to-MP stagnation point distance ratio of 1.29 +/- 0.10. This is near the 1.33 value for the Earth's magnetosphere, implying a similar sunward shape at Saturn. Study of the structure and dynamical behavior of the outer magnetosphere, both in the sunward hemisphere and the magnetotail region using combined plasma and magnetic field data, suggest that Saturn's magnetosphere is more similar to that of Earth than that of Jupiter

Building dependency graph for slicing erlang programs

Program slicing is a well-known technique that utilizes dependency graphs and static program analysis. Our goal is to perform impact analysis of Erlang programs based on the resulted program slices, that is we want to measure the impact of any change made on the source code: especially we want to select a subset of test cases which must be rerun after the modification. However impact analyzer tools exist for object oriented languages, the used dependency graphs heavily depend on the syntax and semantics of the used programming language, thus we introduce dependency graphs for a dynamically typed functional programming language, Erlang

Isomorphism test for digraphs with weighted edges

Colour refinement is at the heart of all the most efficient graph isomorphism software packages. In this paper we present a method for extending the applicability of refinement algorithms to directed graphs with weighted edges. We use Traces as a reference software, but the proposed solution is easily transferrable to any other refinement-based graph isomorphism tool in the literature. We substantiate the claim that the performances of the original algorithm remain substantially unchanged by showing experiments for some classes of benchmark graphs