Before revision:  As one of our main results, we devise an algorithm for d-Hitting Set that runs in time n^{O(d^2 + ( d/\epsilon )) }  , uses O(  d^2 + ( d/\epsilon) log n) bits of space, and achieves an approximation ratio of O((d/ \epsilon ) n^{\epsilon}) for any positive \epsilon \leq 1 and any constant d\in N . In particular, this yields a factor-O( d log n) approximation algorithm which  uses O(log^2 n) bits of space  .
After revision:  As one of our main results, we devise an algorithm for d-Hitting Set that runs in time n^{O(d^2 + ( d/\epsilon )) } , uses O( d^2 + ( d/\epsilon) log n) bits of space, and achieves an approximation ratio of O((d/ \epsilon ) n^{\epsilon}) for any positive \epsilon \leq 1 and any constant d\in N . In particular, this yields a factor-O( d log n) approximation algorithm which uses O(log^2 n) bits of space (for constant d) .