Material Symmetry to Partition Endgame Tables

Many games display some kind of material symmetry . That is, some sets of game elements can be exchanged for another set of game elements, so that the resulting position will be equivalent to the original one, no matter how the elements were arranged on the board. Material symmetry is routinely used in card game engines when they normalize their internal representation of the cards. Other games such as chinese dark chess also feature some form of material symmetry, but it is much less clear what the normal form of a position should be. We propose a principled approach to detect material symmetry. Our approach is generic and is based on solving multiple rel- atively small sub-graph isomorphism problems. We show how it can be applied to chinese dark chess , dominoes , and skat . In the latter case, the mappings we obtain are equivalent to the ones resulting from the standard normalization process. In the two former cases, we show that the material symmetry allows for impressive savings in memory requirements when building endgame tables. We also show that those savings are relatively independent of the representation of the tables

GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs

We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale graphs and complex networks are particularly considered in its design. The problems currently supported include maximum clique, graph colouring, maximum independent set, minimum vertex clique covering, minimum dominating set, as well as the longest simple cycle problem. Suboptimal solutions and intervals for optimal objective values are estimated using scalable heuristics. The tool is designed with extensibility in mind, with the view of further problems and both new fast and high-performance heuristics to be added in the future. GraphCombEx has already been successfully used as a support tool in a number of recent research studies using combinatorial optimisation to analyse complex networks, indicating its promise as a research software tool

High-Frequency Dynamics of Ocean pH: A Multi-Ecosystem Comparison

The effect of Ocean Acidification (OA) on marine biota is quasi-predictable at best. While perturbation studies, in the form of incubations under elevated pCO2, reveal sensitivities and responses of individual species, one missing link in the OA story results from a chronic lack of pH data specific to a given species' natural habitat. Here, we present a compilation of continuous, high-resolution time series of upper ocean pH, collected using autonomous sensors, over a variety of ecosystems ranging from polar to tropical, open-ocean to coastal, kelp forest to coral reef. These observations reveal a continuum of month-long pH variability with standard deviations from 0.004 to 0.277 and ranges spanning 0.024 to 1.430 pH units. The nature of the observed variability was also highly site-dependent, with characteristic diel, semi-diurnal, and stochastic patterns of varying amplitudes. These biome-specific pH signatures disclose current levels of exposure to both high and low dissolved CO2, often demonstrating that resident organisms are already experiencing pH regimes that are not predicted until 2100. Our data provide a first step toward crystallizing the biophysical link between environmental history of pH exposure and physiological resilience of marine organisms to fluctuations in seawater CO2. Knowledge of this spatial and temporal variation in seawater chemistry allows us to improve the design of OA experiments: we can test organisms with a priori expectations of their tolerance guardrails, based on their natural range of exposure. Such hypothesis-testing will provide a deeper understanding of the effects of OA. Both intuitively simple to understand and powerfully informative, these and similar comparative time series can help guide management efforts to identify areas of marine habitat that can serve as refugia to acidification as well as areas that are particularly vulnerable to future ocean change

Solving the Sequential Ordering Problem with Automatically Generated Lower Bounds

The Sequential Ordering Problem (SOP) is a version of the Asymmetric Traveling Salesman Problem (ATSP) where precedence constraints on the vertices must also be observed. The SOP has many real life applications and it has proved to be a great challenge (there are SOPs with 40-50 vertices which have not been solved optimally yet with significant computational effort). We use novel branch&bound search algorithms with lower bounds obtained from homomorphic abstractions of the original state space. Our method is asymptotically optimal. In one instance, it has proved a solution value to be optimal for an open problem while it also has matched best known solutions quickly for many unsolved problems from the TSPLIB. Our method of deriving lower bounds is general and applies to other variants of constrained ATSPs as well

Retrograde Analysis Of Patterns Versus Metaprogramming

this paper) alternatively colors an empty hexagon. It is played on an hexagonal grid as shown in Figure 1. It is usually played on 11x11 or larger sizes, and the game is currently solved until the size 7x7. The goal of the game is to connect two opposite borders with hexagons of the same color, each border is given a pre-defined color, and the player that has this color tries to connect the two borders of his color. The players, Red and Blue, take turns putting hexagons of their color on empty hexagon