Faster width-dependent algorithm for mixed packing and covering LPs
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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+ approximate solution in time O(Nw/ ), where N is number of nonzero entries in the constraint matrix and w is the maximum number of nonzeros in any constraint. This run-time is better than Nesterov's smoothing algorithm which requires O(N√(n)w/ ) 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 while breaking the infamous ℓ_∞ barrier to eliminate the factor of √(n). The current best width-independent algorithm for this problem runs in time O(N/^2) [Young-arXiv-14] and hence has worse running time dependence on . Many real life instances of the mixed packing-covering problems exhibit small width and for such cases, our algorithm can report higher precision results when compared to width-independent algorithms. As a special case of our result, we report a 1+ approximation algorithm for the densest subgraph problem which runs in time O(md/ ), where m is the number of edges in the graph and d is the maximum graph degree.
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