The Complexity of Kings

A king in a directed graph is a node from which each node in the graph can be reached via paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king [Lan53]. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of the semifeasible sets [HNP98,HT05] and in the study of the complexity of reachability problems [Tan01,NT02]. In this paper, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is known to belong to $\Pi_2^p$ [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is $\Pi_2^p$-complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is $\Pi_2^p$-complete. We also obtain $\Pi_2^p$-completeness results for k-kings in succinctly specified j-partite tournaments, $k,j \geq 2$, and we generalize our main construction to show that $\Pi_2^p$-completeness holds for testing k-kingship in succinctly specified families of tournaments for all $k \geq 2$

When You Come at the King You Best Not Miss

A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu [Jian Shen et al., 2003] in investigating the query complexity of finding a king, that is, the number of arcs in T one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least ?(n^{4/3}) arcs and provided a strategy that queries at most O(n^{3/2}). While this upper bound has not yet been improved for the original problem, [Biswas et al., 2017] proved that with O(n^{4/3}) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n^{4/3} polylog n) queries, we can identify a (1/2+2/17)-king. To achieve this goal we use a novel structural result for tournaments

Qui plus fait, miex vault : evaluating combat in Malory's Le Morte d'Arthur

"I seek to correct a deficiency in Malory studies, the inadequate attention paid to the thematic implications of Malory's treatment of armor and various forms of combat. I explore three topics: Malory's changes to his source texts in his depiction of the style of armor and the form of tournaments insofar as they bear on defining the temporal setting of his narrative; Malory's implicit definition of the virtues of the ideal knight errant and tournament champion; and his definition of the knight-commander in war. My over-arching conclusion is that Malory's treatment of combat is essentially ethical. The style of both armor and tournament combat in Malory both suggest a pre-fifteenth-century temporal setting, a finding which suggests that scholars' attempts to evaluate the behavior of Malory's knights in the light of fifteenth-century tournament regulations are anachronistic. "checklist" of virtues for the ideal knight is created through discussions by Malory's leading knights. Malory's "score-keeping" of tournament performance is therefore important to clarify which knights are qualified to participate in this discussion. The chivalric virtues recommended conform closely to those praised in the medieval manuals of Lull, Charny, and Bouvet. The qualities so defined become the standard by which to judge the relative merits of Malory's knights. From a close comparison and contrast of leading knights, Lancelot emerges as the ideal exemplar of the knight errant and tourney champion. From a similar comparative study of knights who take command in war, in particular, King Arthur, Lancelot, Tristram, and Gawain, Lancelot also emerges as the ideal knight-commander, both in the light of the skills of military leadership recommended by medieval military manuals as well as the ethical standard provided by the medieval concept of the just war."--Abstract from author supplied metadata

Employee Compensation: Research and Practice

[Excerpt] An organization has the potential to remain viable only so long as its members choose to participate and engage in necessary role behaviors (March & Simon, 1958; Katz & Kahn, 1966). To elicit these contributions, an organization must provide inducements that are of value to its members. This exchange or transaction process is at the core of the employment relationship and can be viewed as a type of contract, explicit or implicit, that imposes reciprocal obligations on the parties (Barnard, 1936; Simon, 1951; Williamson, 1975; Rousseau, 1990). At the heart of that exchange are decisions by employers and employees regarding compensation

April 27, 2000

The Breeze is the student newspaper of James Madison University in Harrisonburg, Virginia

Strategies for the page and strategies for the stage. The interplay of image and language

RESUMEN: Este estudio se centra en ciertos rasgos comunes de las estrategias utilizadas en los libros de emblemas y en los dramas concebidos para su representación en los tablados del Renacimiento inglés. Estos rasgos incluyen tanto la pantomima por señas expresiva como la enigmática, las intervenciones del coro, retablos alegóricos, y los usos de sentencias en el discurso dramático. PALABRAS CLAVES: Shakespeare, pantomima por señas, intervenciones del coro, retablos alegóricos, sentencias. ABSTRACT: The present study dwells on certain common features of the strategies followed by emblem books and by drama composed for the stages of the English Renaissance. These features include the expressive as well as the enigmatic dumb show, choric interventions, allegorical tableaux, and the uses of sententiæ in dramatic discourse. KEYWORDS: Shakespeare; Dumb Show; Choric Interventions; Allegorical Tableaux; Sentential.