On the Monadic Second-Order Transduction Hierarchy

We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder where we set C \subseteq K if, and only if, there exists a transduction {\tau} such that C\subseteq{\tau}(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type {\omega}+3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n \in N, of all paths, of all trees, and of all grids

On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

In the \emph{tollbooth problem}, we are given a tree \bT=(V,E) with $n$ edges, and a set of $m$ customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and the objective is to price the edges of \bT such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the \emph{highway problem}, is when \bT is restricted to be a line. For the tollbooth problem, we present a randomized $O(\log n)$-approximation, improving on the current best $O(\log m)$-approximation. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of \bT. In this case, we present an algorithm that returns a $(1-\epsilon)$-approximation, for any $\epsilon > 0$, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard. Finally, we show that in the \emph{coupon model}, when we allow some items to be priced below zero to improve the overall profit, the problem becomes even APX-hard

Dyck Paths, Motzkin Paths and Traffic Jams

It has recently been observed that the normalization of a one-dimensional out-of-equilibrium model, the Asymmetric Exclusion Process (ASEP) with random sequential dynamics, is exactly equivalent to the partition function of a two-dimensional lattice path model of one-transit walks, or equivalently Dyck paths. This explains the applicability of the Lee-Yang theory of partition function zeros to the ASEP normalization. In this paper we consider the exact solution of the parallel-update ASEP, a special case of the Nagel-Schreckenberg model for traffic flow, in which the ASEP phase transitions can be intepreted as jamming transitions, and find that Lee-Yang theory still applies. We show that the parallel-update ASEP normalization can be expressed as one of several equivalent two-dimensional lattice path problems involving weighted Dyck or Motzkin paths. We introduce the notion of thermodynamic equivalence for such paths and show that the robustness of the general form of the ASEP phase diagram under various update dynamics is a consequence of this thermodynamic equivalence.Comment: Version accepted for publicatio

On graph combinatorics to improve eigenvector-based measures of centrality in directed networks

Producción CientíficaWe present a combinatorial study on the rearrangement of links in the structure of directed networks for the purpose of improving the valuation of a vertex or group of vertices as established by an eigenvector-based centrality measure. We build our topological classification starting from unidirectional rooted trees and up to more complex hierarchical structures such as acyclic digraphs, bidirectional and cyclical rooted trees (obtained by closing cycles on unidirectional trees). We analyze different modifications on the structure of these networks and study their effect on the valuation given by the eigenvector-based scoring functions, with particular focus on α-centrality and PageRank.Ministerio de Economía, Industria y Competitividad (project TIN2014-57226-P)Generalitat de Catalunya (project SGR2014- 890)Ministerio de Ciencia, Innovación y Universidades (project MTM2012-36917-C03-01

On the Parameterized Intractability of Monadic Second-Order Logic

One of Courcelle's celebrated results states that if C is a class of graphs of bounded tree-width, then model-checking for monadic second order logic (MSO_2) is fixed-parameter tractable (fpt) on C by linear time parameterized algorithms, where the parameter is the tree-width plus the size of the formula. An immediate question is whether this is best possible or whether the result can be extended to classes of unbounded tree-width. In this paper we show that in terms of tree-width, the theorem cannot be extended much further. More specifically, we show that if C is a class of graphs which is closed under colourings and satisfies certain constructibility conditions and is such that the tree-width of C is not bounded by \log^{84} n then MSO_2-model checking is not fpt unless SAT can be solved in sub-exponential time. If the tree-width of C is not poly-logarithmically bounded, then MSO_2-model checking is not fpt unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time

Graph homomorphisms between trees

In this paper we study several problems concerning the number of homomorphisms of trees. We give an algorithm for the number of homomorphisms from a tree to any graph by the Transfer-matrix method. By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees. These applications include a far reaching generalization of Bollob\'as and Tyomkyn's result concerning the number of walks in trees. Some other highlights of the paper are the following. Denote by $\hom(H,G)$ the number of homomorphisms from a graph $H$ to a graph $G$. For any tree $T_m$ on $m$ vertices we give a general lower bound for $\hom(T_m,G)$ by certain entropies of Markov chains defined on the graph $G$. As a particular case, we show that for any graph $G$, $$\exp(H_{\lambda}(G))\lambda^{m-1}\leq\hom(T_m,G),$$ where $\lambda$ is the largest eigenvalue of the adjacency matrix of $G$ and $H_{\lambda}(G)$ is a certain constant depending only on $G$ which we call the spectral entropy of $G$. In the particular case when $G$ is the path $P_n$ on $n$ vertices, we prove that $$\hom(P_m,P_n)\leq \hom(T_m,P_n)\leq \hom(S_m,P_n),$$ where $T_m$ is any tree on $m$ vertices, and $P_m$ and $S_m$ denote the path and star on $m$ vertices, respectively. We also show that if $T_m$ is any fixed tree and $$\hom(T_m,P_n)>\hom(T_m,T_n),$$ for some tree $T_n$ on $n$ vertices, then $T_n$ must be the tree obtained from a path $P_{n-1}$ by attaching a pendant vertex to the second vertex of $P_{n-1}$. All the results together enable us to show that |\End(P_m)|\leq|\End(T_m)|\leq|\End(S_m)|, where \End(T_m) is the set of all endomorphisms of $T_m$ (homomorphisms from $T_m$ to itself).Comment: 47 pages, 15 figure

Location routing problems on trees

This paper addresses combined location/routing problems defined on trees, where not all vertices have to be necessarily visited. A mathematical programming formulation is presented, which has the integrality property. The formulation models a directed forest where each connected component hosts at least one open facility, which becomes the root of the component. The problems considered can also be optimally solved with ad-hoc solution algorithms. Greedy type optimal algorithms are presented for the cases when all vertices have to be visited and facilities have no set-up costs. Facilities set-up costs can be handled with low-order interchanges, whose optimality check is a re-statement of the complementary slackness conditions of the proposed formulation. The general problems where not all vertices have to be necessarily visited can also be optimally solved with low-order optimal algorithms based on recursions.Peer ReviewedPostprint (author's final draft

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems