Minimize makespan on identical machines when jobs are vectors
This is the Vector Scheduling (VS) problem. Unless P = NP, VS admits no constant factor approximation algorithm when $d$ is part of the input: 

Chekuri, C., Khanna, S.: On multidimensional packing problems.
SIAM J. Comput., 33(4):837-851, 2004. 

The same paper gives a polynomial-time approximation scheme when $d$ is not part of the input, and a polynomial time $O((\log d)^2)$ approximation when $d$ is part of the input. The latter was improved to $O(\log d)$ by Meyerson, Roytman and Tagiku (APPROX 2013). For more references and recent results you can refer to:

Bansal, N., Vredeveld, T., van der Zwaan, R.: Approximating Vector Scheduling: Almost Matching Upper and Lower Bounds. Lecture Notes in Computer Science, 8392:47-59, 2014.