A fast approximation scheme for fractional covering problems with variable upper bounds
Lisa Fleischer · 2004
We present the first combinatorial approximation scheme that yields a pure approximation guarantee for linear programs that are either covering problems with upper bounds on variables, or their duals. Existing approximation schemes for mixed covering and packing problems do not simultaneously satisfy packing and covering constraints exactly. We present the first combinatorial approximation scheme that returns solutions that simultaneously satisfy general positive covering constraints and upper bounds on variable values. For input parameter ɛ> 0, the returned solution has positive linear objective function value at most 1+ɛ times the optimal value. The general algorithm requires O(ɛ 2 m log(c T u)) iterations, where c is the objective cost vector, u is the vector of upper bound values, and m is the number of variables. Each iteration uses an oracle that finds an (approximately) most violated constraint. A natural set of problems that our work addresses are linear programs for various network design problems: generalized Steiner network, vertex connectivity, directed connectivity, capacitated network design, group Steiner forest. The integer versions of these problems are all NP-hard. For each of them, there is an approximation algorithm that rounds the solution to the corresponding linear program relaxation. If the LP solution is not feasible, then the corresponding integer solution will also not be feasible. Solving the linear program is often the computational bottleneck in these problems, and thus a fast approximation scheme for the LP relaxation means faster approximation algorithms. For these applications, we introduce a new modification of the push-relabel maximum flow algorithm that allows us to perform each iteration in amortized O(|E|+|V | log |V |) time, instead of one maximum flow per iteration that is implied by the straight forward adaptation of our general algorithm. In conjunction with an observation that reduces the number of iterations to |E | log |V | for {0, 1} constraint matrices, the modification allows us to obtain an algorithm that is faster than existing exact or approximate algorithms by a factor of at least O(|E|) and by a factor of O(|E | log |V |) if the number of demand pairs is Ω(|V |). 1