On the Inapproximability of Disjoint Paths and Minimum Steiner Forest with Bandwidth Constraints

In this paper, we study the inapproximability of several well known optimization problems in network optimization. We show that the max directed vertex-disjoint paths problem cannot be approximated within ratio 2log1 n unless NP DTIME[2polylog n], the max directed edge-disjoint paths problem cannot be approximated within ratio 2log1 n unless NP DTIME[2polylog n], the integer multicommodity ow problem in directed graph cannot be approximated within ratio 2log1 n unless NP DTIME[2polylog n], the max undirected vertex-disjoint paths problem does not have a polynomial time approximation scheme unless P = NP , and the minimum Steiner forest with bandwidth constraints problem cannot be approximated within ratio exp(poly(n)) unless P=NP.