Approximating sparse covering integer programs online

Abstract

A covering integer program (CIP) is a mathematical program of the form min{c⊤x | Ax ≥ 1,0 ≤ x ≤ u, x ∈ ℤn}, where all entries in A, c , u are nonnegative. In the online setting, the constraints (i.e., the rows of the constraint matrix A) arrive over time, and the algorithm can only increase the coordinates of x to maintain feasibility. As an intermediate step, we consider solving the covering linear program (CLP) online, where the integrality constraints are dropped. Our main results are (a) an O(log k)-competitive online algorithm for solving the CLP, and (b) an O(log k·logℓ)-competitive randomized online algorithm for solving the CIP. Here k ≤ n and ℓ ≤ m respectively denote the maximum number of nonzero entries in any row and column of the constraint matrix A. Our algorithm is based on the online primal-dual paradigm, where a novel ingredient is to allow dual variables to increase and decrease throughout the course of the algorithm. It is known that this result is the best possible for polynomial-time online algorithms, even in the special case of set cover (where all entries in A, c , and u are 0 or 1.