Faster width-dependent algorithm for mixed packing and covering LPs

In this paper, we give a faster width-dependent algorithm for mixed packing-covering LPs. Mixed packing-covering LPs are fundamental to combinatorial optimization in computer science and operations research. Our algorithm finds a 1 \eps approximate solution in time O(Nw/ \varepsilon), where N is number of nonzero entries in the constraint matrix, and w is the maximum number of nonzeros in any constraint. This algorithm is faster than Nesterov's smoothing algorithm which requires O(N\sqrt{n}w/ \eps) time, where n is the dimension of the problem. Our work utilizes the framework of area convexity introduced in [Sherman-FOCS'17] to obtain the best dependence on \varepsilon while breaking the infamous \ell_{\infty} barrier to eliminate the factor of \sqrt{n} .