Flooding Overcomes Small Covering Constraints

The paper concerns covering with submodular cost (CSC), a class of optimization problems of the form min {c(x) : x in Rn+; (forall i) x in Si}, where the cost c : Rn+ -> R+ is any increasing submodular function and each constraint set Si is any monotone subset of Rn+. CSC generalizes mixed integer linear covering programs with box constraints (MICP): min {c.x : Ax >= b; x <= u; x in Rn+; (forall j in I) x_j in Z}, where A has non-negative entries. The paper presents a d-approximation algorithm for CSC, where d is the maximum number of variables on which any constraint depends. The basic algorithm is a simple greedy algorithm: repeatedly choose any unmet constraint and invest equally in all its variables until the constraint is met. Specific implementations include: (i) A nearly linear-time d-approximation algorithm for MICP. (ii) The first on-line d-competitive algorithms for CSC and MICP, including a memoryless randomized algorithm. (On-line CSC includes on-line set cover, rent-or-buy, paging, weighted caching/paging, and file/web caching; the on-line algorithms here generalize classic k-competitive on-line algorithms including LANDLORD and HARMONIC.) (iii) The first sublinear-time distributed d-approximation algorithms for CSC and MICP, requiring O(log n) and O(log^2 n) rounds (w.h.p.) for the case d=2 and the general case, respectively. The approach is similar to the local-ratio method, but deals naturally with variables over arbitrary non-negative domains.