We study the approximability of two natural NP-hard problems. The first problem is congestion minimization in directed networks. In this problem, we are given a directed graph and a set of source-sink pairs. The goal is to route all the pairs with minimum congestion on the network edges. The second problem is machine scheduling, where we are given a set of jobs, and for each job, there is a list of intervals on which it can be scheduled. The goal is to find the smallest number of machines on

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... ch all jobs can be scheduled such that no two jobs overlap in their execution on any machine. 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 Max SNP hardness was known for each problem. We make progress in closing this gap by showing that both problems are Ω(log log n)-hard to approximate unless NP ⊆ DTIME(n O(log log log n) ).