Inapproximability of Edge-Disjoint Paths and low congestion routing on undirected graphs

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 ɛ > 0 there exists an α > 0 such that for 1 ⩽ c ⩽ $$ \frac{{\alpha log log V }} {{\log \log \log V}} $$ , 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 $$ {1 \mathord{\left/ {\vphantom {1 {\left( {\log V} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\log V} \right)}}^{\frac{{1 - \varepsilon }} {{c + 2}}} $$ fraction of the terminal pairs, even if we allow congestion c. This implies a $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 2}}} $$ hardness of approximation for EDPwC and an Ω(log logV/log log logV) hardness of approximation for the undirected congestion minimization problem. These results hold assuming NP ⊊ ∪ d 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 $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 1}}} $$ . Note that by setting c=1 this implies that the regular EDP problem is $$ \left( {\log V} \right)^{\frac{1} {2} - \varepsilon } $$ 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 $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 1}}} $$ 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.