Greedy D{\ensuremath{\Delta}}-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a greedy ${\ensuremath{\Delta}}$-approximation algorithm for monotone covering , a generalization of many fundamental NP-hard covering problems. The approximation ratio ${\ensuremath{\Delta}}$ is the maximum number of variables on which any constraint depends. (For example, for vertex cover, ${\ensuremath{\Delta}}$ is 2.) The algorithm unifies, generalizes, and improves many previous algorithms for fundamental covering problems such as vertex cover, set cover, facilities location, and integer and mixed-integer covering linear programs with upper bound on the variables. The algorithm is also the first ${\ensuremath{\Delta}}$-competitive algorithm for online monotone covering, which generalizes online versions of the above-mentioned covering problems as well as many fundamental online paging and caching problems. As such it also generalizes many classical online algorithms, including lru, fifo, fwf, balance, greedy-dual, greedy-dual size (a.k.a. landlord ), and algorithms for connection caching, where ${\ensuremath{\Delta}}$ is the cache size. It also gives new ${\ensuremath{\Delta}}$-competitive algorithms for upgradable variants of these problems, which model choosing the caching strategy and an appropriate hardware configuration (cache size, CPU, bus, network, etc.).