Improved Approximation Algorithms and Lower Bounds for Search-Diversification Problems
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We study several questions related to diversifying search results. We give improved approximation algorithms in each of the following problems, together with some lower bounds. - We give a polynomial-time approximation scheme (PTAS) for a diversified search ranking problem [Bansal et al., ICALP 2010] whose objective is to minimizes the discounted cumulative gain. Our PTAS runs in time n^2^O(log(1/ϵ)/ϵ)· m^O(1) where n denotes the number of elements in the databases. Complementing this, we show that no PTAS can run in time f(ϵ) · (nm)^2^o(1/ϵ) assuming Gap-ETH; therefore our running time is nearly tight. Both of our bounds answer open questions of Bansal et al. - We next consider the Max-Sum Dispersion problem, whose objective is to select k out of n elements that maximizes the dispersion, which is defined as the sum of the pairwise distances under a given metric. We give a quasipolynomial-time approximation scheme for the problem which runs in time n^O_ϵ(log n). This improves upon previously known polynomial-time algorithms with approximate ratios 0.5 [Hassin et al., Oper. Res. Lett. 1997; Borodin et al., ACM Trans. Algorithms 2017]. Furthermore, we observe that known reductions rule out approximation schemes that run in n^õ_ϵ(log n) time assuming ETH. - We consider a generalization of Max-Sum Dispersion called Max-Sum Diversification. In addition to the sum of pairwise distance, the objective includes another function f. For monotone submodular f, we give a quasipolynomial-time algorithm with approximation ratio arbitrarily close to (1 - 1/e). This improves upon the best polynomial-time algorithm which has approximation ratio 0.5 by Borodin et al. Furthermore, the (1 - 1/e) factor is tight as achieving better-than-(1 - 1/e) approximation is NP-hard [Feige, J. ACM 1998].
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