Beating Simplex for Fractional Packing and Covering Linear Programs

We give an approximation algorithm for packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of I +epsiv of OPT l+ epsiv of OPT (the optimal cost) in time O(n + (r +c) log(n) / epsiv²). For dense problems (with r,c = O(-radicn)) the time is Omega (n log(n) / epsiv²)-linear even as epsiv rarr 0. In comparison, previous Lagrangian-relaxation algorithms generally take at least Omega(n log(n)/epsiv²) time, while (for small epsiv) the Simplex algorithm typically takes at least Omega(n min(r, c)) time.