Approximation algorithms for variable-sized materials constructing tree-form structures in undirected graph

Abstract

It is known that one-dimensional variable-sized bin packing problem and network optimization problem are classical combinatorial optimization problems. Inspired by this, we consider a new problem of variable-sized materials constructing some tree-form structures in undirected graph, where all edges spliced in such a tree-form structure are assembled from some pieces of m types of materials. More precisely, that is defined as follows: given a weighted graphtree-form structure  and some pieces of m types of materials, where :b E R   is a cost function , we will attempt to construct a subgraph G from G , having the structure  , such that each edge in G is further constructed by given m types of materials , the new objective is to minimize the total cost to construct subgraph G , where the total cost is sum of the cost to purchase materials and the cost to construct all edges in such a subgraph .G For this new problem, we obtained the following two main results.(1)When the structure  is a spanning tree, we design a 2approximation algorithm to solve the problem of variable-sized materials constructing spanning tree. (2) When the structure  is a single-source shortest paths tree, we design a 2approximation algorithm to solve the problem of variablesized materials constructing single-source shortest paths tree.