Approximation algorithms and hardness of integral concurrent flow
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Routing in undirected graphs with constant congestion
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Maximum edge-disjoint paths in k-sums of graphs
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
| Hi-index | 0.00 |
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 ⊊ ∪dZPTIME($$2^{\log ^{d_n } }$$). 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.