On the Approximability of Multistage Min-Sum Set Cover
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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 (π^0, π^1, …, π^T) on n elements, based on a sequence of requests (R^1, …, R^T). We aim to minimize the total cost of updating π^t-1 to π^t, quantified by the Kendall tau distance D_KT(π^t-1, π^t), plus the total cost of covering each request R^t with the current permutation π^t, quantified by the position of the first element of R^t in π^t. Using a reduction from Set Cover, we show that DSSC does not admit an O(1)-approximation, unless P = NP, and that any o(log n) (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(log^2 n) in general instances, by randomized rounding, and within a factor of O(r^2), if all requests have cardinality at most r, by deterministic rounding.
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