Reference:
Petteri Kaski. Algorithms for classification of combinatorial objects. Research Report A94, Helsinki University of Technology, Laboratory for Theoretical Computer Science, Espoo, Finland, June 2005. Doctoral dissertation.
Abstract:
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 elevenround 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 onefactorizations of regular graphs of order 12 for and , a classification of the nearresolutions of 2(13,4,3) designs together with the associated thirteenplayer whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.
Keywords:
classification algorithm, isomorphism, isomorph rejection, nearresolvable design, onefactorization, orderly algorithm, regular graph, resolvable design, Steiner triple system
Suggested BibTeX entry:
@techreport{HUTTCSA94,
address = {Espoo, Finland},
author = {Petteri Kaski},
institution = {Helsinki University of Technology, Laboratory for Theoretical Computer Science},
month = {June},
note = {Doctoral dissertation},
number = {A94},
pages = {56},
title = {Algorithms for Classification of Combinatorial Objects},
type = {Research Report},
year = {2005},
}
