Set Cover and Vertex Cover with Delay
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The set cover problem is one of the most fundamental problems in computer science. We present the problem of online set cover with delay (SCD). A family of sets with costs and a universe of elements are known in advance. Requests then arrive over time on the elements, and each request accumulates delay cost until served by the algorithm through buying a containing set. A request can only be served by sets that are bought after the request's arrival, and thus a set may be bought an unbounded number of times over the course of the algorithm. This property sets SCD apart from previous considerations of set cover in the online setting, in which there are no delays, elements are covered immediately, and sets stay bought permanently. This allows SCD to describe an unbounded process, with an unlimited number of requests for any given universe. For the SCD problem, we show an O( k· n)-competitive randomized algorithm, where n is the number of elements and k is the maximum number of sets containing any single element. We also show a lower bound of Ω(√( k)) and Ω(√( n)) on the competitiveness of any algorithm for SCD. For the special case of Vertex Cover with Delay (VCD), we show a simple 3-competitive deterministic algorithm. The O( k· n)-competitive algorithm is based on exponential weights combined with the max operator (in contrast to most algorithms employing exponential weights, which use summation). The lower bound is described by a recursive construction.
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