Automatic enumeration of regular objects

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer Sequence

The enumeration of planar graphs via Wick's theorem

A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enumeration of the graphs embeddable on a given 2-dimensional surface (a main research topic of contemporary enumerative combinatorics) can also be formulated as the Gaussian matrix integral of an ice-type partition function. Some of the most puzzling conjectures of discrete mathematics are related to the notion of the cycle double cover. We express the number of the graphs with a fixed directed cycle double cover as the Gaussian matrix integral of an Ihara-Selberg-type function.Comment: 23 pages, 2 figure

Asymptotic behavior of the number of Eulerian orientations of graphs

We consider the class of simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). For this class of graphs we determine the asymptotic behavior of the number of Eulerian orientations. In addition, we establish some new properties of the Laplacian matrix, as well as an estimate of a conditionality of matrices with the asymptotic diagonal predominanceComment: arXiv admin note: text overlap with arXiv:1104.304

Level Eulerian Posets

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem--Mahler--Lech theorem, the ${\bf ab}$-series of a level poset is shown to be a rational generating function in the non-commutative variables ${\bf a}$ and ${\bf b}$. In the case the poset is also Eulerian, the analogous result holds for the ${\bf cd}$-series. Using coalgebraic techniques a method is developed to recognize the ${\bf cd}$-series matrix of a level Eulerian poset

A time- and space-optimal algorithm for the many-visits TSP

The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of $n$ cities that visits each city $c$ a prescribed number $k_c$ of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time $n^{O(n)} + O(n^3 \log \sum_c k_c )$ and requires $n^{\Theta(n)}$ space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length $\sum_c k_c$ of the tour, allowing the algorithm to handle instances with very long tours. The \emph{superexponential} dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time $2^{O(n)}$, i.e.\ \emph{single-exponential} in the number of cities, using \emph{polynomial} space. Our algorithm is deterministic, and arguably both simpler and easier to analyse than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.Comment: Small fixes, journal versio