Hardness of routing with congestion in directed graphs

Given as input a directed graph on n vertices and a set ofsource-destination pairs, we study the problem of routing themaximum possible number of source-destination pairs on paths, suchthat at most c(N) paths go through any edge. We show that theproblem is hard to approximate within an NΩ(1/c(N)) factoreven when we compare to the optimal solution that routes pairs onedge-disjoint paths, assuming NP doesn't have NO(log logN)-time randomized algorithms. Here the congestion c(N) can beany function in the range 1 ≤ c(N) ≤ α log N/log log N for some absolute constant α > 0. The hardness result is in the right ballpark since a factor NO(1/c(N)) approximation algorithm is known for this problem, viarounding a natural multicommodity-flow relaxation. We also give asimple integrality gap construction that shows that themulticommodity-flow relaxation has an integrality gap of NΩ(1/c) for c ranging from 1 to Θ((log n)/(log log n)). A solution to the routing problem involves selecting which pairs tobe routed and what paths to assign to each routed pair. Two naturalrestrictions can be placed on input instances to eliminate one ofthese aspects of the problem complexity. The first restriction is toconsider instances with perfect completeness; an optimalsolution is able to route all pairs with congestion 1 in suchinstances. The second restriction to consider is the uniquepaths property where each source-destination pair has a unique pathconnecting it in the instance. An important aspect of our result isthat it holds on instances with any one of these tworestrictions. Our hardness construction with the perfectcompleteness restriction allows us to conclude that the directedcongestion minimization problem, where the goal is to route allpairs with minimum congestion, is hard to approximate to within afactor of Ω(log N/log log N). On the other hand, thehardness construction with unique paths property allows us toconclude an NΩ(1/c) inapproximability bound also for theall-or-nothing flow problem. This is in a sharp contrast to theundirected setting where the all-or-nothing flow problem is known tobe approximable to within a poly-logarithmic factor.