We investigate the polynomial-time approximability of the multistage version
of Min-Sum Set Cover (DSSC), a natural and intriguing generalization
of the classical List Update problem. In DSSC, we maintain a
sequence of permutations (π0,π1,…,πT) on n elements, based
on a sequence of requests (R1,…,RT). We aim to minimize the total
cost of updating πt−1 to πt, quantified by the Kendall tau
distance DKT(πt−1,πt), plus the total cost of
covering each request Rt with the current permutation πt, quantified by
the position of the first element of Rt in πt.
Using a reduction from Set Cover, we show that DSSC does not admit
an O(1)-approximation, unless P=NP, and that any
o(logn) (resp. o(r)) approximation to DSSC implies a
sublogarithmic (resp. o(r)) approximation to Set Cover (resp. where each
element appears at most r times). Our main technical contribution is to show
that DSSC can be approximated in polynomial-time within a factor of
O(log2n) in general instances, by randomized rounding, and within a factor
of O(r2), if all requests have cardinality at most r, by deterministic
rounding