Regular combinatorial maps

The classical approach to maps is by cell decomposition of a surface. A combinatorial map is a graph-theoretic generalization of a map on a surface. Besides maps on orientable and non-orientable surfaces, combinatorial maps include tessellations, hypermaps, higher dimensional analogues of maps, and certain toroidal complexes of Coxeter, Shephard, and Grünbaum. In a previous paper the incidence structure, diagram, and underlying topological space of a combinatorial map were investigated. This paper treats highly symmetric combinatorial maps. With regularity defined in terms of the automorphism group, necessary and sufficient conditions for a combinatorial map to be regular are given both graph- and group-theoretically. A classification of regular combinatorial maps on closed simply connected manifolds generalizes the well-known classification of metrically regular polytopes. On any closed manifold with nonzero Euler characteristic there are at most finitely many regular combinatorial maps. However, it is shown that, for nearly any diagram D, there are infinitely many regular combinatorial maps with diagram D. A necessary and sufficient condition for the regularity of rank 3 combinatorial maps is given in terms of Coxeter groups. This condition reveals the difficulty in classifying the regular maps on surfaces. In light of this difficulty an algorithm for generating a large class of regular combinatorial maps that are obtained as cyclic coverings of a given regular combinatorial map is given.     

Combinatorial sums through Riordan arrays

The aim of this work is to show how Riordan arrays are able to generate and close combinatorial identities, by means of the method of coefficients (generating functions). We also show how the same approach can be used to deal with other combinatorial problems, for instance asymptotic approximation and combinatorial inversion. Finally, we propose a method for generating new combinatorial sums by extending the concept of Riordan arrays to bi-infinite matrices.     

An asymptotical study of combinatorial optimization problems by means of statistical mechanics

The analogy between combinatorial optimization and statistical mechanics has proven to be a fruitful object of study. Simulated annealing, a metaheuristic for combinatorial optimization problems, is based on this analogy. In this paper we show how a statistical mechanics formalism can be utilized to analyze the asymptotic behavior of combinatorial optimization problems with sum objective function and provide an alternative proof for the following result: Under a certain combinatorial condition and some natural probabilistic assumptions on the coefficients of the problem, the ratio between the optimal solution and an arbitrary feasible solution tends to one almost surely, as the size of the problem tends to infinity, so that the problem of optimization becomes trivial in some sense. Whereas this result can also be proven by purely probabilistic techniques, the above approach allows one to understand why the assumed combinatorial condition is essential for such a type of asymptotic behavior.     

Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension

Motivated by the definition of combinatorial scalar curvature given by Cooper and Rivin, we introduce a new combinatorial scalar curvature. Then we define the discrete quasi-Einstein metric, which is a combinatorial analogue of the constant scalar curvature metric in smooth case. We find that discrete quasi-Einstein metric is critical point of both the combinatorial Yamabe functional and the quadratic energy functional we defined on triangulated 3-manifolds. We introduce combinatorial curvature flows, including a new type of combinatorial Yamabe flow, to study the discrete quasi-Einstein metrics and prove that the flows produce solutions converging to discrete quasi-Einstein metrics if the initial normalized quadratic energy is small enough. As a corollary, we prove that nonsingular solution of the combinatorial Yamabe flow with nonpositive initial curvatures converges to discrete quasi-Einstein metric. The proof relies on a careful analysis of the discrete dual-Laplacian, which we interpret as the Jacobian matrix of curvature map.     

A combinatorial proof of a fixed point property

A class of finite simplicial complexes, called pseudo cones, is developed that has a number of useful combinatorial properties. A partially ordered set is a pseudo cone if its order complex is a pseudo cone. Pseudo cones can be constructed from other pseudo cones in a number of ways. Pseudo cone ordered sets include finite dismantlable ordered sets and finite truncated noncomplemented lattices. The main result of the paper is a combinatorial proof of the fixed simplex property for finite pseudo cones in which a combinatorial structure is constructed that relates fixed simplices to one another. This gives combinatorial proofs of some well known non-constructive results in the fixed point theory of finite partially ordered sets.     

The semigroup of combinatorial configurations

We elaborate on the existence and construction of the so-called combinatorial configurations. The main result is that for fixed degrees the existence of such configurations is given by a numerical semigroup. The proof is constructive giving a method to obtain combinatorial configurations with parameters large enough.     

Multilevel Refinement for Combinatorial Optimisation Problems

We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at each level. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid techniques). However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying process and extract some generic guidelines for its future use on other combinatorial problems.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K_λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK^-1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

Toward a model for students’ combinatorial thinking

Combinatorics has many applications in different disciplines, however, only a few studies have explored students’ combinatorial thinking at the upper secondary and tertiary levels concurrently. The present research is a grounded theory study of eight Year 12 and five undergraduate students, who have participated in semi-structured interviews and responded to eight combinatorial tasks. Three types of combinatorial tasks were designed: combinatorial reasoning, evaluating, and problem-posing tasks. In the open coding phase of data analysis, seventy-one codes were identified which categorized into seven main categories at the axial coding phase. At the selective coding phase, five relationships between categories were identified that led to designing a model of students’ combinatorial thinking. The model consists of three movements: Horizontal, vertical downward, and vertical upward movement. It is asserted that this model could be used to improve the quality of teaching combinatorics, and also as a lens to explore students’ combinatorial thinking.     

Embedding graphs in surfaces

The basic topological facts about closed curves in $\text{R}$² and triangulability of surfaces are used to prove the folk theorem that any surface embedding of a graph is combinatorial. A basic technical lemma for this proof (a version of what it means to apply scissors to an embedded graph) is then used to give a rigourous definition of the combinatorial boundary of a face and also to introduce a combinatorial definition of equivalence of embeddings. This latter definition is on the one hand easily seen to correspond correctly to the natural topological notion of equivalence, and on the other hand to give equivalence classes in 1-1 correspondence with the classes coming from combinatorial definitions of earlier authors.     

Configuration spaces,bistellar moves,and combinatorial formulae for the first Pontryagin class

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based on the notion of a universal local formula introduced by the author and on the usage of bistellar moves. We give a brief sketch for the first formula and a rather detailed exposition for the second one. For the second formula, we also succeed to simplify it by providing a new simpler algorithm for decomposing a cycle in the graph of bistellar moves of two-dimensional combinatorial spheres into a linear combination of elementary cycles.     

Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture

We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the diameters of polyhedral graphs. One particular variant has a diameter which satisfies the best known upper bound on the diameters of polyhedra. Another variant has superlinear asymptotic diameter, and together with some combinatorial operations, gives a concrete approach for disproving the Linear Hirsch Conjecture.     

The imbedding index of a graph

The natural extension of MacLane's combinatorial approach to planar imbeddings is seen to yield a combinatorial formulation of imbedding of a graph in a pseudosurface. This leads to a combinatorially defined parameter for all graphs, called the imbedding index. A generalization of the Heaword inequality is then proved for this parameter.     

On the combinatorial structure of chromatic scheduling polytopes

Chromatic scheduling polytopes arise as solution sets of the bandwidth allocation problem in certain radio access networks, supplying wireless access to voice/data communication networks for customers with individual communication demands. To maintain the links, only frequencies from a certain spectrum can be used, which typically causes capacity problems. Hence it is necessary to reuse frequencies but no interference must be caused by this reuse. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard, and there do not even exist polynomial time algorithms with a fixed quality guarantee.As algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems, the goal is to apply such methods to the bandwidth allocation problem. For that, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring the combinatorial structure of chromatic scheduling polytopes for increasing frequency spans. We observe that the polytopes pass through various stages—emptyness, non-emptyness but low-dimensionality, full-dimensionality but combinatorial instability, and combinatorial stability—as the frequency span increases. We discuss the thresholds for this increasing “quantity” giving rise to a new combinatorial “quality” of the polytopes, and we prove bounds on these thresholds. In particular, we prove combinatorial equivalence of chromatic scheduling polytopes for large frequency spans and we establish relations to the linear ordering polytope.     

An integrable semi-discrete equation and combinatorial numbers with their combinatorial interpretations

A Hankel type determinant solution for an integrable semi-discrete equation is presented. As an application, the relations between the solution and combinatorial numbers are discussed, which lead to new combinatorial numbers. The so-called Motzkin-like numbers are obtained, and the corresponding combinatorial interpretation is given. Additionally, it is also shown that some lattice paths have connections with the special solution.     

Recurrence relation with two indices and plane compositions

The aim of this paper is to study analytical and combinatorial methods to solve a special type of recurrence relation with two indices. It is shown that the recurrence relation enumerates a natural combinatorial object called a plane composition. In addition, further interesting recurrence relations arise in the study of statistics for these plane compositions.     

Investigating undergraduate students’ proof schemes and perspectives about combinatorial proof

Combinatorics is an area of mathematics with accessible, rich problems and applications in a variety of fields. Combinatorial proof is an important topic within combinatorics that has received relatively little attention within the mathematics education community, and there is much to investigate about how students reason about and engage with combinatorial proof. In this paper, we use Harel and Sowder’s (1998) proof schemes to investigate ways that students may characterize combinatorial proofs as different from other types of proof. We gave five upper-division mathematics students combinatorial-proof tasks and asked them to reflect on their activity and combinatorial proof more generally. We found that the students used several of Harel and Sowder’s proof schemes to characterize combinatorial proof, and we discuss whether and how other proof schemes may emerge for students engaging in combinatorial proof. We conclude by discussing implications and avenues for future research.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K _λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK ^?1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

A market-based multi-agent system model for decentralized multi-project scheduling

We consider a multi-project scheduling problem, where each project is composed of a set of activities, with precedence relations, requiring specific amounts of local and shared (among projects) resources. The aim is to complete all the project activities, satisfying precedence and resource constraints, and minimizing each project schedule length. The decision making process is supposed to be decentralized, with as many local decision makers as the projects. A multi-agent system model, and an iterative combinatorial auction mechanism for the agent coordination are proposed. We provide a dynamic programming formulation for the combinatorial auction problem, and heuristic algorithms for both the combinatorial auction and the bidding process. An experimental analysis on the whole multi-agent system model is discussed.     

Combinatorial optimization with interaction costs: Complexity and solvable cases

We introduce and study the combinatorial optimization problem with interaction costs (COPIC). COPIC is the problem of finding two combinatorial structures, one from each of two given families, such that the sum of their independent linear costs and the interaction costs between elements of the two selected structures is minimized. COPIC generalizes the quadratic assignment problem and many other well studied combinatorial optimization problems, and hence covers many real world applications. We show how various topics from different areas in the literature can be formulated as special cases of COPIC. The main contributions of this paper are results on the computational complexity and approximability of COPIC for different families of combinatorial structures (e.g. spanning trees, paths, matroids), and special structures of the interaction costs. More specifically, we analyze the complexity if the interaction cost matrix is parameterized by its rank and if it is a diagonal matrix. Also, we determine the structure of the intersection cost matrix, such that COPIC is equivalent to independently solving linear optimization problems for the two given families of combinatorial structures.