The Full Degree Spanning Tree Problem

Given a graph G, we study the problem of finding a spanning tree T that maximizes the number of vertices of full degree; that is, the number of vertices whose degree in T equals its degree in G. We prove a few general bounds and then analyze this parameter on various classes of graphs including grid graphs, hypercubes, and random regular graphs. We also explore a related problem that focuses on maximizing the number of leaves in a spanning tree of a graph

On the Directed Full Degree Spanning Tree Problem

Abstract. We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning out-tree T of D such that at least k vertices in T have the same out-degree as in D. We show that this problem is W[1]-hard even on the class of directed acyclic graphs. In the dual version, called Reduced Degree Spanning Tree, one is required to construct a spanning out-tree T such that at most k vertices in T have out-degrees that are different from that in D. We show that this problem is fixed-parameter tractable and that it admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k 2 ) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with running time O(5.942 k · n O(1) ), where n is the number of vertices in the input digraph

Models and Algorithms for Some Covering Problems on Graphs

2014 - 2015Several real-life problems as well as problems of theoretical importance within the field of Operations Research are combinatorial in nature. Combinatorial Optimization deals with decision-making problems defined on a discrete space. Out of a finite or countably infinite set of feasible solutions, one has to choose the best one according to an objective function. Many of these problems can be modeled on undirected or directed graphs. Some of the most important problems studied in this area include the Minimum Spanning Tree Problem, the Traveling Salesman Problem, the Vehicle Routing Problem, the Matching Problem, the Maximum Flow Problem. Some combinatorial optimization problems have been modeled on colored (labeled) graphs. The colors can be associated to the vertices as well as to the edges of the graph, depending on the problem. The Minimum Labeling Spanning Tree Problem and the Minimum Labeling Hamiltonian Cycle Problem are two examples of problems defined on edge-colored graphs. Combinatorial optimization problems can be divided into two groups, according to their complexity. The problems that are easy to solve, i.e. problems polynomially solvable, and those that are hard, i.e. for which no polynomial time algorithm exists. Many of the well-known combinatorial optimization problems defined on graphs are hard problems in general. However, if we know more about the structure of the graph, the problems can become more tractable. In some cases, they can even be shown to be polynomial-time solvable. This particularly holds for trees...[edited by Author]XIV n.s

The Full Degree Spanning Tree Problem

The full degree spanning tree problem is defined as follows: given a connected graph G = (V; E) find a spanning tree T so as to maximize the number of vertices whose degree in T is the same as in G (these are called vertices of "full" degree). We show that this problem is NP-hard. We also present almost optimal approximation algorithms for it assuming coR 6= NP . For the case of general graphs our approximation factor is \Theta( p n). Using Hastad's result on the hardness of approximating clique, we can show that if there is a polynomial time approximation algorithm for our problem with a factor of O(n 1 2 \Gammaffl ) then coR = NP . For the case of planar graphs, we present a polynomial time approximation scheme. Additionally, we present some experimental results comparing our algorithm to the previous heuristic used for this problem