Given a universe U of n elements and a weighted collectionS of m subsets of U, the universal set cover problem is to a-priori map each element u \u2208U to a set S(u) \u2208 S containing u, so that X \u2286 U is covered by S(X) = \u222au\u2208XS(u). The aim is finding a mapping such that the cost of S(X) is as close as possible to the optimal set-cover cost for X. (Such problems are also called oblivious or a-priori optimization problems.) Unfortunately, for every universal mapping, the cost of S(X) can be \u03a9(\u221a n) times larger than optimal if the set X is adversarially chosen.<\/p>\n
In this paper we study the performance on average, when X is a set of randomly chosen elements from the universe: we show how to efficiently find a universal map whose expected cost is O(logmn) times the expected optimal cost. In fact, we give a slightly improved analysis and show that this is the best possible. We generalize these ideas to weighted set cover and show similar guarantees to (non-metric) facility location, where we have to balance the facility opening cost with the cost of connecting clients to the facilities. We show applications of our results to universal multi-cut and disc-covering problems, and show how all these universal mappings give us stochastic online algorithms with the same competitive factors.<\/p>\n","protected":false},"excerpt":{"rendered":"
Given a universe U of n elements and a weighted collectionS of m subsets of U, the universal set cover problem is to a-priori map each element u \u2208U to a set S(u) \u2208 S containing u, so that X \u2286 U is covered by S(X) = \u222au\u2208XS(u). The aim is finding a mapping such […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"49th Annual IEEE Symposium on Foundations of Computer Science","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"347\u2013356","msr_page_range_start":"347","msr_page_range_end":"356","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"49th Annual IEEE Symposium on Foundations of Computer 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