Lower Bound

In epidemiological models of infectious diseases the basic reproduction number ${\mathcal{R}_0}$ is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for ${\mathcal{R}_0}$ can be obtained from the digraphs in the reduction process.

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, these bounds almost match the known upper bounds. The results are applied to improve the lower bounds for other related problems such as Maximum Induced Matching and Maximum Leaf Spanning Tree.

This research is motivated by issues faced by a large manufacturer of semiconductor devices. Semiconductor manufacturing companies allocate millions of dollars every year for new types of machine tools for their facilities. Typically these are special purpose machine tools which are made to order. The rate of change in products and technology makes it diÅcult for manufacturers to have a

In this paper we give lower bounds for the minimum distance of evaluation codes constructed from complete intersections in toric varieties. This generalizes the results of Gold-Little-Schenck and Ballico-Fontanari who considered evaluation codes on complete intersections in the projective space.