Abstract:
In the undirected Edge-Disjoint Paths problem with Congestion (EDPwC), we are given an undirected graph with $V$ nodes, a set of terminal pairs and an integer $c$. The objective is to route as many terminal pairs as possible, subject to the constraint that at most $c$ demands can be routed through any edge in the graph. When $c=1$, the problem is simply referred to as the Edge-Disjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs. Our main result is that for every $\ve>0$ there exists an $\alpha>0$ such that for $1\le c\le \frac{\alpha\log\log V}{\log\log\log V}$, it is hard to distinguish between instances where we can route all terminal pairs on edge-disjoint paths, and instances where we can route at most a $1/(\log V)^{\frac{1-\ve}{c+2}}$ fraction of the terminal pairs, even if we allow congestion $c$. This implies a $(\log V)^{\frac{1-\ve}{c+2}}$ hardness of approximation for EDPwC and an $\Omega(\log\log V/\log\log\log V)$ hardness of approximation for the undirected congestion minimization problem. These results hold assuming ${\rm NP} \not\subseteq \cup_d {\rm ZPTIME}(2^{\log^d n})$. In the case that we do not require perfect completeness, i.e.\ we do not require that all terminal pairs are routed for ``yes-instances'', we can obtain a slightly better inapproximability ratio of $(\log V)^{\frac{1-\ve}{c+1}}$. Note that by setting $c=1$ this implies that the regular EDP problem is $(\log V)^{\frac{1}{2}-\ve}$ hard to approximate. Using standard reductions, our results extend to the node-disjoint versions of these problems as well as to the directed setting. We also show a $(\log V)^{\frac{1-\ve}{c+1}}$ inapproximability ratio for the All-or-Nothing Flow with Congestion (ANFwC) problem, a relaxation of EDPwC, in which the flow unit routed between the source-sink pairs does not have to follow a single path, so the resulting flow is not necessarily integral.