Randomized rounding: a technique for provably good algorithms and algorithmic proofs
Combinatorica - Theory of Computing
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
SIAM Journal on Computing
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
A unified approach to approximating resource allocation and scheduling
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Approximating the Throughput of Multiple Machines in Real-Time Scheduling
SIAM Journal on Computing
The Approximability of Constraint Satisfaction Problems
SIAM Journal on Computing
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Interval selection: applications, algorithms, and lower bounds
Journal of Algorithms
Polylogarithmic inapproximability
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
A new multilayered PCP and the hardness of hypergraph vertex cover
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Constraint Satisfaction: The Approximability of Minimization Problems
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
Approximation Algorithms for the Job Interval Selection Problem and Related Scheduling Problems
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
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 congestion minimization problem
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Guest column: inapproximability results via Long Code based PCPs
ACM SIGACT News
Logarithmic hardness of the directed congestion minimization problem
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The NP-completeness column: The many limits on approximation
ACM Transactions on Algorithms (TALG)
Hardness of routing with congestion in directed graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On the approximation of the single source k-splittable flow problem
Journal of Discrete Algorithms
Almost-tight hardness of directed congestion minimization
Journal of the ACM (JACM)
Convex combinations of single source unsplittable flows
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Edge Disjoint Paths in Moderately Connected Graphs
SIAM Journal on Computing
Edge disjoint paths in moderately connected graphs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
WAOA'04 Proceedings of the Second international conference on Approximation and Online Algorithms
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We study the approximability of two natural NP-hard problems. The first problem is congestion minimization in directed networks. We are given a directed capacitated graph and a set of source-sink pairs. The goal is to route all pairs with minimum congestion on the network edges. A special well-studied case of this problem is the edge-disjoint paths problem, where all edges have unit capacities. The second problem is discrete machine scheduling, where we are given a set of jobs, and for each job a list of intervals in which it can be scheduled. The goal is to find the smallest number of machines on which all jobs can be scheduled, such that no two jobs assigned to the same machine overlap. Both problems are known to be O(log n/log log n)-approximable via the randomized rounding technique of Raghavan and Thompson. However, until recently, only a Max SNP hardness was known for each problem. We make some progress in closing this gap by showing that both problem are Ω(log log n)-hard to approximate unless NP ⊆ DTIME(nO(log log log n)). Our hardness proof for congestion minimization holds even for the special case of the edge-disjoint paths problem.