The recent breakthrough paper by Calude et al. has given the first algorithm for solving parity games in quasi-polynomial time, where previously the best algorithms were mildly subexponential. We devise an alternative...
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ISBN:
(纸本)9781509030187
The recent breakthrough paper by Calude et al. has given the first algorithm for solving parity games in quasi-polynomial time, where previously the best algorithms were mildly subexponential. We devise an alternative quasi-polynomial time algorithm based on progress measures, which allows us to reduce the space required from quasi-polynomial to nearly linear. Our key technical tools are a novel concept of ordered tree coding, and a succinct tree coding result that we prove using bounded adaptive multi-counters, both of which are interesting in their own right.
Recent work has shown that the classical framework of solving optimization problems by obtaining a fractional solution to a linear program (LP) and rounding it to an integer solution can be extended to the online sett...
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ISBN:
(纸本)9781611972511
Recent work has shown that the classical framework of solving optimization problems by obtaining a fractional solution to a linear program (LP) and rounding it to an integer solution can be extended to the online setting using primal-dual techniques. The success of this new framework for online optimization can be gauged from the fact that it has led to progress in several longstanding open questions. However, to the best of our knowledge, this framework has previously been applied to LPs containing only packing or only covering constraints, or minor variants of these. We extend this framework in a fundamental way by demonstrating that it can be used to solve mixed packing and covering LPs online, where packing constraints are given of-fine and covering constraints are received online. The objective is to minimize the maximum multiplicative factor by which any packing constraint is violated, while satisfying the covering constraints. Our results represent the first algorithm that obtains a polylogarithmic competitive ratio for solving mixed LPs online. We then consider two canonical examples of mixed LPs: unrelated machine scheduling with startup costs, and capacity constrained facility location. We use ideas generated from our result for mixed packing and covering to obtain polylogarithmic-competitive algorithms for these problems. We also give lower bounds to show that the competitive ratios of our algorithms are nearly tight.
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