Algorithms for classification of combinatorial objects

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one game in every round. Such a characterization is called a classification for the objects of interest. Classification is typically conducted up to a notion of structural equivalence (isomorphism) between the objects. For example, one can view two tournament schedules as having the same structure if one can be obtained from the other by renaming the teams and reordering the rounds. This thesis examines algorithms for classification of combinatorial objects up to isomorphism. The thesis consists of five articles – each devoted to a specific family of objects – together with a summary surveying related research and emphasizing the underlying common concepts and techniques, such as backtrack search, isomorphism (viewed through group actions), symmetry, isomorph rejection, and computing isomorphism. From an algorithmic viewpoint the focus of the thesis is practical, with interest on algorithms that perform well in practice and yield new classification results; theoretical properties such as the asymptotic resource usage of the algorithms are not considered. The main result of this thesis is a classification of the Steiner triple systems of order 19. The other results obtained include the nonexistence of a resolvable 2-(15, 5, 4) design, a classification of the one-factorizations of k-regular graphs of order 12 for k ≤ 6 and k = 10, 11, a classification of the near-resolutions of 2-(13, 4, 3) designs together with the associated thirteen-player whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.reviewe

The Near Resolvable 2-(13, 4, 3) Designs and Thirteen-Player Whist Tournaments

A v-player whist tournament is a schedule of games, where in each round the v players are partitioned into games of four players each with at most one player left over. In each game two of the players play as partners against the other two. All pairs of players must play in the same game exactly three times during the tournament; of those three times, they are to play as partners exactly once. Whist tournaments for v players are known to exist for all v ≡ 0, 1 (mod 4). The special cases of directed whist tournaments and triplewhist tournaments are known to exist for all sufficiently large v, but for small v several open cases remain. In this paper we introduce a correspondence between near resolvable 2-(v, k, λ) designs and a particular class of codes. The near resolvable 2-(13, 4, 3) designs are classified by classifying the corresponding codes with an orderly algorithm. Finally, the thirteen-player whist tournaments are enumerated starting from the near resolvable 2-(13, 4, 3) designs

Modern Political Philosophy

Openly licensed anthology focused on the theme of Modern Political Philosophy. Contains Leviathan by Thomas Hobbes; The Prince by Niccolò Machiavelli; The Communist Manifesto by Friedrich Engels and Karl Marx; The Eighteenth Brumaire of Louis Bonaparte by Karl Marx ; On the Jewish Question by Karl Marx Economic & Philosophic Manuscripts of 1844 by Karl Marx; The German Ideology by Karl Marx; Capital by Karl Marx ; Second Treatise of Government by John Locke; The Social Contract & Discourses by Jean-Jacques Rousseau; Beyond Good and Evil by Friedrich Wilhelm Nietzsch