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In a sweep cover problem, mobile sensors move around to collect information from positions of interest (PoIs) periodically and timely. A PoI is sweep-covered if it is visited at least once in every time period t. In this paper, we study approximation algorithms on three types of sweep cover problems. The partial sweep cover problem (PSC) aims to use the minimum number of mobile sensors to sweep-cover at least a given number of PoIs. The prize-collecting sweep cover problem aims to minimize the cost of mobile sensors plus the penalties on those PoIs that are not sweep-covered. The budgeted sweep cover problem (BSC) aims to use a budgeted number N of mobile sensors to sweep-cover as many PoIs as possible. We propose a unified approach which can yield approximation algorithms for PSC and PCSC within approximation ratio at most 8, and a bicriteria (4, 1 2 )-approximation algorithm for BSC (that is, no more than 4N mobile sensors are used to sweep-cover at least 1 2 opt PoIs, where opt is the number of PoIs that can be sweep-covered by an optimal solution). Furthermore, our results for PSC and BSC can be extended to their weighted version, and our algorithm for PCSC answers a question proposed in Liang etal. (Theor Comput Sci, 2022) on PCSC 

Given an element set E of order n, a collection of subsets [Formula: see text], a cost c S on each set [Formula: see text], a covering requirement r e for each element [Formula: see text], and an integer k, the goal of a minimum partial set multicover problem (MinPSMC) is to find a subcollection [Formula: see text] to fully cover at least k elements such that the cost of [Formula: see text] is as small as possible and element e is fully covered by [Formula: see text] if it belongs to at least r e sets of [Formula: see text]. This problem generalizes the minimum k-union problem (MinkU) and is believed not to admit a subpolynomial approximation ratio. In this paper, we present a [Formula: see text]-approximation algorithm for MinPSMC, in which [Formula: see text] is the maximum size of a set in S. And when [Formula: see text], we present a bicriteria algorithm fully covering at least [Formula: see text] elements with approximation ratio [Formula: see text], where [Formula: see text] is a fixed number. These results are obtained by studying the minimum density subcollection problem with (or without) cardinality constraint, which might be of interest by itself.