Stochastic analyses for online combinatorial optimization problems

In this paper, we study online algorithms when the input is not chosen adversarially, but consists of draws from some given probability distribution. While this model has been studied for online problems like paging and <i>k</i>-server, it is not known how to beat the Φ(log <i>n</i>) bound for online Steiner tree if at each time instant, the demand vertex is a uniformly random vertex from the graph. For the online Steiner tree problem, we show that if each demand vertex is an independent draw from some probability distribution π: <i>V</i> → [0, 1], a variant of the natural greedy algorithm achieves <b>E</b><sub>ω</sub>[<b><i>A</i></b>(ω)]/<b>E</b><sub>ω</sub>[OPT (ω)] = <i>O</i>(1); moreover, this result can be extended to some other subadditive problems. Both assumptions that the input sequence consists of <i>independent</i> draws from π, and that π is known to the algorithm are both essential; we show (almost) logarithmic lower bounds if either assumption is violated. Moreover, we give preliminary results on extending the Steiner tree results above to the related "expected ratio" measure <b>E</b><sub>ω</sub>[ω(ω)/OPT (ω)]. Finally, we use these ideas to give an average-case analysis of the Universal TSP problem.

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