Improved local computation algorithm for set cover via sparsification

• Main Authors: Mitrović, Slobodan (Author), Rubinfeld, Ronitt (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor), Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory (Contributor)
• Format: Article
• Language: English
• Published: Society for Industrial and Applied Mathematics, 2021-04-08T16:45:50Z.
• Subjects: Article
• Online Access: Get fulltext

We design a Local Computation Algorithm (LCA) for the set cover problem. Given a set system where each set has size at most s and each element is contained in at most t sets, the algorithm reports whether a given set is in some fixed set cover whose expected size is O(log s) times the minimum fractional set cover value. Our algorithm requires s (log s t (log s·(log log s+log log t)) queries. This result improves upon the application of the reduction of [Parnas and Ron, TCS'07] on the result of [Kuhn et al., SODA'06], which leads to a query complexity of (st) (log s·log t . To obtain this result, we design a parallel set cover algorithm that admits an efficient simulation in the LCA model by using a sparsification technique introduced in [Ghaffari and Uitto, SODA'19] for the maximal independent set problem. The parallel algorithm adds a random subset of the sets to the solution in a style similar to the PRAM algorithm of [Berger et al., FOCS'89]. However, our algorithm differs in the way that it never revokes its decisions, which results in a fewer number of adaptive rounds. This requires a novel approximation analysis which might be of independent interest. O ) O O )
Swiss National Science Foundation (Grant P2ELP2_181772)
MIT-IBM Watson AI Lab (Rsearch Collaboration Agreement W1771646)
National Science Foundation (U.S.) (Awards CCF-1740751, CCF-1733808, and IIS-1741137)