Partial Subliner Time Approximation and Inapproximation for Maximum Coverage

We develop a randomized approximation algorithm for the classical maximum coverage problem, which given a list of sets $A_1,A_2,\cdots, A_m$ and integer parameter $k$, select $k$ sets $A_{i_1}, A_{i_2},\cdots, A_{i_k}$ for maximum union $A_{i_1}\cup A_{i_2}\cup\cdots\cup A_{i_k}$. In our algorithm, each input set $A_i$ is a black box that can provide its size $|A_i|$, generate a random element of $A_i$, and answer the membership query $(x\in A_i?)$ in $O(1)$ time. Our algorithm gives $(1-{1\over e})$-approximation for maximum coverage problem in $O(p(m))$ time, which is independent of the sizes of the input sets. No existing $O(p(m)n^{1-\epsilon})$ time $(1-{1\over e})$-approximation algorithm for the maximum coverage has been found for any function $p(m)$ that only depends on the number of sets, where $n=\max(|A_1|,\cdots,| A_m|)$ (the largest size of input sets). The notion of partial sublinear time algorithm is introduced. For a computational problem with input size controlled by two parameters $n$ and $m$, a partial sublinear time algorithm for it runs in a $O(p(m)n^{1-\epsilon})$ time or $O(q(n)m^{1-\epsilon})$ time. The maximum coverage has a partial sublinear time $O(p(m))$ constant factor approximation. On the other hand, we show that the maximum coverage problem has no partial sublinear $O(q(n)m^{1-\epsilon})$ time constant factor approximation algorithm. It separates the partial sublinear time computation from the conventional sublinear time computation by disproving the existence of sublinear time approximation algorithm for the maximum coverage problem.