Covering a graph by nontrivial paths

Abstract

El objetivo de esta tesis es estudiar problemas sobre coberturas de grafos por caminos. Sea G un grafo y sea P un conjunto de caminos vértice-disjuntos en G. Decimos que P es una cobertura por caminos si cada vértice de G pertenece a un camino en p. En el problema de la cobertura mínima por caminos, deseamos encontrar una cobertura cuya cardinalidad sea mínima. En este problema, el conjunto P puede contener caminos triviales (de un solo vértice). Estudiamos el problema de encontrar una cobertura compuesta por caminos no triviales. Consideramos el problema de la existencia de tal cobertura y mostramos cómo reducirlo a un problema de emparejamiento. Además, consideramos los problemas de minimización y maximización correspondientes. Proponemos formulaciones lineales enteras para estos problemas y presentamos algunos resultados experimentales. Para el problema de maximización, mostramos que se puede resolver en tiempo polinomial. Finalmente, también consideramos una versión del problema en grafos con pesos en las aristas. En estos casos buscamos una cobertura cuyo peso total sea mínimo o máximo. Demostramos el primer problema es polinomial, y el segundo es NP-difícil. Además, para el caso de maximización, mostramos un algoritmo de aproximación de factor constante.

In this thesis our aim is to study problems concerning path covers of graphs. Let G be a graph and let P be a set of pairwise vertex-disjoint paths in G. We say that P is a path cover if every vertex of G belongs to a path in P. In the minimum path cover problem, one wishes to find a path cover of minimum cardinality. In this problem, known to be NP-hard, the set P may contain trivial (single-vertex) paths. We study the problem of finding a path cover composed only of nontrivial paths. First, we consider the corresponding existence problem, and show how to reduce it to a matching problem. From this reduction, we derive a characterization that allows us to find, in polynomial time, a certificate for both the YES-answer and the NO-answer. When trivial paths are forbidden, for the feasible instances, one may consider either minimizing or maximizing the number of paths in the cover. We show that the minimization problem on feasible instances is computationally equivalent to the minimum path cover problem: their optimum values coincide and they have the same approximation threshold. Moreover, we propose integer linear formulations for these problems and present some experimental results. For the maximization problem, we show that it can be solved in polynomial time. Finally, we also consider a weighted version of the path cover problem, in which we seek for a path cover with minimum or maximum total weight (the number of paths does not matter), and we show that while the first is polynomial, the second is NP-hard. Furthermore, for the maximization case, we show a constant-factor approximation algorithm. We also show that, when the input graph has bounded treewidth, both problems can be solved in linear time. To conclude, we present an integer linear formulation for the case of minimum total weight, and study the polytope obtained when the integrality constraint is relaxed. We show that there are graphs for which this polytope has fractional vertices, and we exhibit some classes of inequalities that are valid for the integral polytope and separate these vertices.