Randomized rounding: a technique for provably good algorithms and algorithmic proofs
Combinatorica - Theory of Computing
Balls and bins: a study in negative dependence
Random Structures & Algorithms
Some optimal inapproximability results
Journal of the ACM (JACM)
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Improved approximations for edge-disjoint paths, unsplittable flow, and related routing problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Approximation algorithms for disjoint paths problems
Approximation algorithms for disjoint paths problems
Journal of Computer and System Sciences
Graph decomposition and a greedy algorithm for edge-disjoint paths
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
New hardness results for congestion minimization and machine scheduling
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
The all-or-nothing multicommodity flow problem
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Multicommodity flow, well-linked terminals, and routing problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Hardness of the undirected edge-disjoint paths problem
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Hardness of the undirected congestion minimization problem
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Hardness of the Undirected Edge-Disjoint Paths Problem with Congestion
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Logarithmic hardness of the directed congestion minimization problem
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Polynomial flow-cut gaps and hardness of directed cut problems
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Almost-tight hardness of directed congestion minimization
Journal of the ACM (JACM)
Polynomial flow-cut gaps and hardness of directed cut problems
Journal of the ACM (JACM)
Unsplittable Flow in Paths and Trees and Column-Restricted Packing Integer Programs
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Truthful unsplittable flow for large capacity networks
ACM Transactions on Algorithms (TALG)
Maximum flows on disjoint paths
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Edge Disjoint Paths in Moderately Connected Graphs
SIAM Journal on Computing
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Routing in undirected graphs with constant congestion
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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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.