On the approximability and exact algorithms for vector domination and related problems in graphs

We consider two graph optimization problems called vector domination and total vector domination. In vector domination one seeks a small subset S of vertices of a graph such that any vertex outside S has a prescribed number of neighbors in S. In total vector domination, the requirement is extended to all vertices of the graph. We prove that these problems (and several variants thereof) cannot be approximated to within a factor of clnn, where c is a suitable constant and n is the number of the vertices, unless P = NP. We also show that two natural greedy strategies have approximation factors ln D+O(1), where D is the maximum degree of the input graph. We also provide exact polynomial time algorithms for several classes of graphs. Our results extend, improve, and unify several results previously known in the literature.Comment: In the version published in DAM, weaker lower bounds for vector domination and total vector domination were stated. Being these problems generalization of domination and total domination, the lower bounds of 0.2267 ln n and (1-epsilon) ln n clearly hold for both problems, unless P = NP or NP \subseteq DTIME(n^{O(log log n)}), respectively. The claims are corrected in the present versio

A Survey on Alliances and Related Parameters in Graphs

In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global $r$-alliances in graphs.We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

Entry and Competition in Local Hospital Markets

There has been considerable consolidation in the hospital industry in recent years. Over 900 deals occurred from 1994-2000, and many local markets, even in large urban areas, have been reduced to monopolies, duopolies, or triopolies. This surge in consolidation has led to concern about its effect on competition in local markets for hospital services. In this paper we examine the impact of market structure on competition in local hospital markets -- specifically, does competition increase with the number of firms? We extend the entry model developed by Bresnahan and Reiss to make use of quantity information, and apply it to data on the U.S. hospital industry. The results from the estimation are striking. In the hospital markets we examine, entry leads to markets becoming competitive quickly. Entry reduces variable profits and increases quantity. Indeed, most of the effects of entry come from having a second and possibly a third firm enter the market. The use of quantity information allows us to infer that entry is consumer welfare increasing.

Spatially resolved texture analysis of Napoleonic War era copper bolts

The spatial resolution achievable by a time-of-flight neutron strain scanner has been harnessed using a new data analysis methodology (NyRTex) to determine, nondestructively, the spatial variation of crystallographic texture in objects of cultural heritage. Previous studies on the crystallographic texture at the centre of three Napoleonic War era copper bolts, which demonstrated the value of this technique in differentiating between the different production processes of the different types of bolts, were extended to four copper bolts from the wrecks of HMS Impregnable (completed 1786), HMS Amethyst (1799), HMS Pomone (1805) and HMS Maeander (1840) along with a cylindrical `segment' of a further incomplete bolt from HMS Pomone. These included bolts with works stamps, allowing comparison with documentary accounts of the manufacturing processes used, and the results demonstrated unequivocally that bolts with a `Westwood and Collins' patent stamp were made using the Collins rather than the Westwood process. In some bolts there was a pronounced variation in texture across the cross section. In some cases this is consistent with what is known of the types of hot and cold working used, but the results from the latest study might also suggest that, even in the mature phase of this technology, some hand finishing was sometimes necessary. This examination of bolts from a wider range of dates is an important step in increasing our understanding of the introduction and evolution of copper fastenings in Royal Navy warships

Alliance free sets in Cartesian product graphs

Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$, and vertex $v\in V$, let $\delta_S(v)=|\{u\in S:uv\in E\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$. Consider the following condition: {equation}\label{alliancecondition} \delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\bar{S}$. A set $S\subseteq V$ that satisfies Condition (\ref{alliancecondition}) for every vertex $v \in S$ is called a \emph{defensive} $k$-\emph{alliance}; for every vertex $v$ in the neighborhood of $S$ is called an \emph{offensive} $k$-\emph{alliance}. A subset of vertices $S\subseteq V$, is a \emph{powerful} $k$-\emph{alliance} if it is both a defensive $k$-alliance and an offensive $(k +2)$-alliance. Moreover, a subset $X\subset V$ is a defensive (an offensive or a powerful) $k$-alliance free set if $X$ does not contain any defensive (offensive or powerful, respectively) $k$-alliance. In this article we study the relationships between defensive (offensive, powerful) $k$-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) $k$-alliance free sets in the factor graphs

Local majorities, coalitions and monopolies in graphs: a review

AbstractThis paper provides an overview of recent developments concerning the process of local majority voting in graphs, and its basic properties, from graph theoretic and algorithmic standpoints