Minimizing normalized volumes of valuations

Star Valuations and Dual Mixed Volumes

Since its creation by Brunn and Minkowski, what has become known as the Brunn Minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data. The machinery of the Brunn Minkowski theory includes mixed volumes (of Minkowski), symmetrization techniques (such as those of Steiner and Blaschke), isoperimetric inequalities (such as the Brunn Mink...

Normalized Tenacity and Normalized Toughness of Graphs

In this paper, we introduce the novel parameters indicating Normalized Tenacity ($T_N$) and Normalized Toughness ($t_N$) by a modification on existing Tenacity and Toughness parameters.  Using these new parameters enables the graphs with different orders be comparable with each other regarding their vulnerabilities. These parameters are reviewed and discussed for some special graphs as well.

Extremal Sets Minimizing Dimension-Normalized Boundary in Hamming Graphs

Extremal Sets Minimizing Dimension-normalized Boundary in Hamming Graphs∗

We prove that the set of first k vertices of a Hamming graph in reverse-lexicographic order constitutes an extremal set minimizing the dimension-normalized edge-boundary over all kvertex subsets of the graph. This generalizes a result of Lindsey and can be used to prove a tight lower bound for the isoperimetric number and the bisection width of arrays.

A note on minimizing the normalized sum of squared workload deviations on m parallel processors

In a fundamental paper on workload balancing on m identical parallel machines, Ho et al. (2009) proved that a schedule which minimizes the normalized sum of squares is necessarily a makespan-optimal one. Based on their theoretical result, they proposed a heuristic algorithm to solve the workload balancing problem. In this note we provide a counter-example for the above-mentioned result and expl...