Combinatorics

Problems and algorithms for covering arrays

by Alan Hartman

Software and Hardware Testing Using Combinatorial Covering Suites

by Alan Hartman

The existence of resolvable Steiner quadruple systems

by Alan Hartman

Misyurov D.A. Dialectical formulas based on the binary notation as the development formulas // Credo New. 2012. №2

by Dmitry Misyurov

The article suggests dialectical formulas based on the binary notation as the development formulas: formula with... more The article suggests dialectical formulas based on the binary notation as the development formulas: formula with dominant and the non-dominant elements; universal formula; formula with symbolic weight of elements; tautological formula. For example, it suggests an opportunity to use the dialectical formulas for modeling and artificial intelligence creation, etc.

Finding a cluster-tilting object for a representation finite cluster-tilted algebra

by Marco Angel Bertani-Økland

Co-authored with Steffen Oppermann and Anette Wrålsen. Colloq. Math., 121(2): 249--263, 2010.

We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in... more We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case.

Solving the 3D Containership Stowage Loading Planning Problem by Representation by Rules and Beam Search

by Anibal Azevedo

Published in  1st International Conference on Operations Research and Enterprise Systems - ICORES 2012

This paper formulates the 3D Container ship Loading Planning Problem (3D CLPP) and also proposes a
new and... more This paper formulates the 3D Container ship Loading Planning Problem (3D CLPP) and also proposes a
new and compact representation to efficiently solve it. Containers on board a Container ship are placed in
vertical stacks, located in different sections. The only way to access the containers is through the top of the
stack. In order to unload a container at a given port  j, it is necessary to remove the container whose
destination is the port j+1, because it is located above the container we want to download. This operation is
called “shifting”. A ship container carrying cargo to several ports may require a large number of shifting
operations. These operations spend a lot of time and cost and can be avoided by using efficient stowage
planning. The key objective of the stowage planning is to minimize the number of container movements and
also the ship instability. The binary formulation of this problem is properly described and also an alternative
formulation called representation by rules is proposed. A Beam Search is combined with representation by
rules to solve the 3D CLPP in manner that ensures that every solution analyzed in the optimization process
is compact and feasible.

On the Algorithmic Proofs of the Four Color Theorem

by Ibrahim Cahit

This paper describes algorithmic proofs of the four color theorem
based on spiral chains. This paper describes algorithmic proofs of the four color theorem
based on spiral chains.

A New Technique for Text Data Compression

by Udita Katugampola

Infinite primitive directed graphs

by Simon Smith

Appeared in the J. Algebraic Combin., (2010)

doi: 10.1007/s10801-009-0190-3

A graph X has connectivity one if it is connected and there exists a vertex the removal of which leaves X... more A graph X has connectivity one if it is connected and there exists a vertex the removal of which leaves X disconnected. If X has connectivity one, a lobe of X is a connected subgraph that is maximal subject to the condition that it does not have connectivity one.

The primitive undirected graphs with connectivity one have been fully classified by Jung and Watkins: the lobes of such graphs are primitive, pairwise-isomorphic and have at least three vertices. When one considers the general case of a directed primitive graph with connectivity one, however, this result no longer holds. In this paper we investigate the structure of these directed graphs, and obtain a complete characterisation.

Orbital graphs of infinite primitive permutation groups

by Simon Smith

Appeared in J. Group Theory, (2007)

doi: 10.1515/JGT.2007.060

If G is a group acting on a set V and a, b are elements of V, the digraph whose vertex set is V and whose arc set is... more If G is a group acting on a set V and a, b are elements of V, the digraph whose vertex set is V and whose arc set is the orbit (a, b)^G is called an orbital digraph of G.

A locally finite digraph X has more than one end if there exists a finite set of vertices Y such that the induced digraph X \ Y contains at least two infinite connected components; if there exists such a set containing precisely one element, then X has connectivity one.

In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivity-one orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterised in another paper (Infinite primitive digraphs) by the author.

Subdegree growth rates of infinite primitive permutation groups

by Simon Smith

Appeared in the J. London Math. Soc., (2010)

doi: 10.1112/jlms/jdq046

A transitive group G of permutations of a set V is primitive if the only G-invariant equivalence relations on V are... more A transitive group G of permutations of a set V is primitive if the only G-invariant equivalence relations on V are the trivial and universal relations. The orbits of a point-stabiliser acting on V are called the suborbits of G and the cardinalities of these suborbits are the subdegrees of G.

If G acts primitively on an infinite set V, and all the suborbits of G are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of G as a non-decreasing sequence, the subdegree growth rates of infinite primitive groups that act distance-transitively on locally finite distance-transitive graphs are extremal, and conjecture that it may be possible to determine the group from the rate of growth.

In this paper it is shown that such an enumeration is not desirable. The examples used to show this provide several novel methods for constructing infinite primitive graphs.
A revised enumeration method is then proposed, and it is shown that, under this, Adeleke and Neumann's question may be answered, at least for groups exhibiting suitable rates of growth.

Rough ends of infinite primitive groups

by Simon Smith

Appeared in the J. Group Theory, (2011)

doi: 10.1515/JGT.2011.108

If G is a group of permutations of a set V, then the suborbits of G are the orbits of point-stabilisers acting on V.... more If G is a group of permutations of a set V, then the suborbits of G are the orbits of point-stabilisers acting on V. The cardinalities of these suborbits are the subdegrees of G. Every infinite primitive permutation group G with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph X with vertex set V, and there is consequently a natural action of G on the ends of X.

We show that if G is closed in the permutation topology of pointwise convergence, then the structure of G is determined by the length of any orbit of G acting on the ends of X.

Distinguishability of infinite groups and graphs

by Simon Smith

Preprint, co-authored with Tom Tucker and Mark Watkins

The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the... more The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the colouring. The distinguishing number of a graph is the distinguishing number of its full automorphism group acting on its vertex set.

We prove that every connected primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any infinite, connected, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number 2. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma.

A classification of primitive permutation groups with finite stabilizers

by Simon Smith

Preprint

We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal... more We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal O'Nan--Scott Theorem to all primitive permutation groups with finite point stabilizers.

Lower bounding edit distances between permutations

by Anthony Labarre

Submitted (2012)

A number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with... more A number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with the connected problems of sorting permutations in "as few moves as possible", using a given set of allowed operations, or computing the number of moves the sorting process requires, often referred to as the \emph{distance} of the permutation. These operations often act on just one or two segments of the permutation, e.g. by reversing one segment or exchanging two segments. The \emph{cycle graph} of the permutation to sort is a fundamental tool in the theory of genome rearrangements, and has proved useful in settling the complexity of many variants of the above problems. In this paper, we present an algebraic reinterpretation of the cycle graph of a permutation $\pi$ as an even permutation $\bar{\pi}$, and show how to reformulate our sorting problems in terms of particular factorisations of the latter permutation. Using our framework, we recover known results in a simple and unified way, and obtain a new lower bound on the \emph{prefix transposition distance} (where a \emph{prefix transposition} displaces the initial segment of a permutation), which is shown to outperform previous results. Moreover, we use our approach to improve the best known lower bound on the \emph{prefix transposition diameter} from $2n/3$ to $\lfloor3n/4\rfloor$, and investigate a few relations between some statistics on $\pi$ and $\bar{\pi}$.

FinalDissertation2010

by Kevin Merges

Feedback vertex sets in tournaments

by Serge Gaspers

Serge Gaspers and Matthias Mnich.
Feedback vertex sets in tournaments.
Journal of Graph Theory.

We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs.
On the... more We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs.
On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740^n minimal feedback vertex sets and that there is an infinite family of tournaments, all having at least 1.5448^n minimal feedback vertex sets. This improves and extends the bounds of Moon (1971).
On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal feedback vertex sets of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum size feedback vertex set in a tournament.

Agents Network on a Grid: An Approach with the Set of Circulant Operators

by Babiga Birregah

The n-Queens Problem in Higher Dimensions

by Shrisha Rao

Co-authored with Jeremiah Barr. Elemente der Mathematik, 61 (4), 2006, pp. 133–137. arXiv:0712.2309v1 [math.CO].

A well-known chessboard problem is that of placing eight queens on the chessboard so that no two queens are able to... more A well-known chessboard problem is that of placing eight queens on the chessboard so that no two queens are able to attack each other. (Recall that a queen can attack anything on the same row, column, or diagonal as itself.) This problem is known to have been studied by Gauss, and can be generalized to an \(n \times n\) board, where \(n \geq 4\). We consider this problem in $d$-dimensional chess spaces, where \(d \geq 3\), and obtain the result that in higher dimensions, $n$ queens do not always suffice (in any arrangement) to attack all board positions. Our methods allow us to obtain the first lower bound
on the number of queens that are necessary to attack all positions in a $d$-dimensional chess space of size $n$, and further to show that for any $k$, there are higher-dimensional chess spaces in which not all positions can be attacked by \(n^k\) queens.

On Independent Sets and Bicliques in Graphs

by Serge Gaspers

Serge Gaspers, Dieter Kratsch, and Mathieu Liedloff.
On independent sets and bicliques in graphs.
Algorithmica.

Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this... more Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner's upper bound on the number of maximal bicliques [Combinatorica, 2000] and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3n/3). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642n), where n is the number of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions.