On a generalisation of the coupon collector problem

Abstract

We consider a generalisation of the classical coupon collector problem. We define a super-coupon to be any $s$-subset of a universe of $n$ coupons. In each round, a random $r$-subset from the universe is drawn and all its $s$-subsets are marked as collected. We show that the time to collect all super-coupons is $\binom{r}{s}^{-1}\binom{n}{s} \log \binom{n}{s}(1 + o(1))$ on average and has a Gumbel limit after a suitable normalisation. In a similar vein, we show that for any $\alpha \in (0, 1)$, the expected time to collect $(1 - \alpha)$ proportion of all super-coupons is $\binom{r}{s}^{-1}\binom{n}{s} \log \big(\frac{1}{\alpha}\big)(1 + o(1))$. The $r = s$ case of this model is equivalent to the classical coupon collector model. We also consider a temporally dependent model where the $r$-subsets are drawn according to the following Markovian dynamics: the $r$-subset at round $k + 1$ is formed by replacing a random coupon from the $r$-subset drawn at round $k$ with another random coupon from outside this $r$-subset. We link the time it takes to collect all super-coupons in the $r = s$ case of this model to the cover time of random walk on a certain finite regular graph and conjecture that in general, it takes $\frac{r}{s} \binom{r}{s}^{-1}\binom{n}{s}\log\binom{n}{s}(1 + o(1))$ time on average to collect all super-coupons.