Sweep　coverage　is　an　important　covering　technique　in　mobile　crowdsensing,　in　which　users　or　participants　are　employed　to　periodically　monitor　a　set　of　points　of　interest　(POIs)　each　with　a　weight　indicating　the　value　of　its　information　to　be　collected.　Traditionally,　each　user　proposes　a　route　along　which　there　is　a　set　of　POIs　to　be　monitored.　The　task　is　to　select　a　set　of　participants　such　that　the　total　weight　of　the　monitored　POIs　is　maximized.　However,　in　real　applications,　users　should　have　the　flexibility　to　offer　several　preferred　routes.　This　arises　our　studied　maximum　sweep　assignment　problem　with　flexibility,　where　each　participant　proposes　several　routes,　and　the　new　task　is　to　strategically　assign　each　participant　a　route　among　their　choices　in　which　the　way　maximizes　the　total　weight　of　the　monitored　POIs.　In　this　paper,　we　first　prove　that　the　problem　is　NP-complete　and　then　devise　two　novel　approximation　algorithms　with　ratios　0　:　5　and　0.632.　Experiments　are　also　conducted　to　evaluate　algorithms＇　practical　performance.　The　results　demonstrate　that　the　proposed　approximate　methods　are　significantly　faster　(with　up　to　two　orders　of　magnitude　runtime　reduction)　than　the　exact　integer　linear　programming　solution.　In　addition,　we　theoretically　study　another　flexible　sweep　coverage　model　in　which　it　costs　to　hire　each　user,　and　the　goal　is　to　cover　all　POIs　multiple　times　(for　more　complete　and　accurate　information)　while　minimizing　the　total　hiring　cost.