Inapproximability proof of DSTLB and USTLB in planar graphs

This document proves the problem of finding a minimum cost Steiner Tree covering k terminals with at most p branching nodes (with outdegree greater than 1), in a directed or an undirected planar graph with n nodes, is hard to approximate within a better ratio than n, even when the parameter p is fixed. 1 Theorem Definition 1. In a undirected (resp. directed) tree, a branching node is a node whose degree (resp. outdegree) is strictly greater than 2 (resp. 1). Problem 1. min-(* , p)-USTLB: Given an undirected graph G = (V, E) with n nodes and a non negative cost function ω on its edges, an integer k and a set X ⊂ V of k terminals, determine, if it exists, a minimum cost tree T * spanning all the nodes of X and containing at most p branching nodes. Problem 2. min-(* , p)-DSTLB: Given a directed graph G = (V, E) with n nodes and a non negative cost function ω on its arcs, a node r, an integer k and a set X ⊂ V of k terminals, determine, if it exists, a minimum cost directed tree T * rooted at r, spanning all the nodes of X and containing at most p branching nodes. Theorem 1. Let < 1 be a real number. If P = NP, the min-(* , p)-DSTLB and the min-(* , p)-USTLB problems in planar graphs with unit costs cannot be approximated within a factor of N where N is the number of nodes in the instance, even if there is a trivial feasible solution. We prove the theorem in the directed case. The proof is similar in the undirected case. Finding a hamiltonian path starting at a specified node v in a 3-connected directed planar graph is a NP-Complete problem [1].