Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension ${rm dim}(X)$ of a comparability graph $X$ is the dimension of any tr
ansitive orientation of X, and by $k$-DIM we denote the class of comparability graphs $X$ with ${rm dim}(X) le k$. It is known that the complements of comparability graphs are exactly function graphs and permutation graphs equal 2-DIM. In this paper, we characterize the automorphism groups of permutation graphs similarly to Jordans characterization for trees (1869). For permutation graphs, there is an extra operation, so there are some extra groups not realized by trees. For $k ge 4$, we show that every finite group can be realized as the automorphism group of some graph in $k$-DIM, and testing graph isomorphism for $k$-DIM is GI-complete.
Power awareness is fast becoming immensely important in computing, ranging from the traditional High Performance Computing applications, to the new generation of data centric workloads. In this work we describe our efforts towards a power efficient
computing paradigm that combines low precision and high precision arithmetic. We showcase our ideas for the widely used kernel of solving systems of linear equations that finds numerous applications in scientific and engineering disciplines as well as in large scale data analytics, statistics and machine learning. Towards this goal we developed tools for the seamless power profiling of applications at a fine grain level. In addition, we verify here previous work on post FLOPS/Watt metrics and show that these can shed much more light in the power/energy profile of important applications.
