A Fast Parallel Implementation of a PTAS for Fractional Packing and Covering Linear Programs

Abstract

We present a parallel implementation of the randomized (1+ε) approximation algorithm for packing and covering linear programs presented by Koufogiannakis and Young (2007). Their approach builds on ideas of the sublinear time algorithm of Grigoriadis and Khachiyan's (Oper Res Lett 18(2):53-58, 1995) and Garg and Köne-mann's (SIAM J Comput 37(2):630-652, 2007) non-uniform-increment amortization scheme. With high probability it computes a feasible primal and dual solution whose costs are within a factor of 1 + ε of the optimal cost. In order to make their algorithm more parallelizable we also implemented a deterministic version of the algorithm, i.e. instead of updating a single random entry at each iteration we updated deterministically many entries at once. This slowed down a single iteration of the algorithm but allowed for larger step-sizes which lead to fewer iterations. We use NVIDIA's parallel computing architecture CUDA for the parallel environment. We report a speedup between one and two orders of magnitude over the times reported by Koufogiannakis and Young (2007).

An FPTAS for the fractional group Steiner tree problem

“…The linear program (2) exhibits the structure of a fractional covering linear program that is more carefully studied later in this paper. Since (2) has exponentially many constraints, state-of-the-art approaches in [19,20,17] cannot be directly applied. Our main task is to adopt their approach and show that it yields a better worst case running time algorithm compared to the previous algorithms, when the number of groups k is large enough.…”