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The paper presents distributed and parallel δ-approximation algorithms for covering problems, where δ is the maximum number of variables on which any constraint depends (for example, δ = 2 for VERTEX COVER). Specific results include the following. ≺ For WEIGHTED VERTEX COVER, the first distributed 2-approximation algorithm taking O(log n) rounds and the first parallel 2-approximation algorithm in RNC. The algorithms generalize to covering mixed integer linear programs (CMIP) with two variables per constraint (δ = 2). ≺ For any covering problem with monotone constraints and submodular cost, a distributed δ-approximation algorithm taking O(log2 |C|) rounds, where |C| is the number of constraints. (Special cases include CMIP, facility location, and probabilistic (two-stage) variants of these problems.)