In the online matching on the line problem, the task is to match a set of requests R online to a given set of servers S. The distance metric between any two points in R ∪ S is a line metric and the objective for the online algorithm is to minimize the sum of distances between matched server-request pairs. This problem is well-studied and -despite recent improvements -there is still a large gap between the best known lower and upper bounds: The best known deterministic algorithm for the problem is O(log 2 n)-competitive, while the best known deterministic lower bound is 9.001. The lower and upper bounds for randomized algorithms are 4.5 and O(log n) respectively.We prove that any deterministic online algorithm which in each round: (i) bases the matching decision only on information local to the current request, and (ii) is symmetric (in the sense that the decision corresponding to the mirror image of some instance I is the mirror image of the decision corresponding to instance I), must be Ω(log n)-competitive. We then extend the result by showing that it also holds when relaxing the symmetry property so that the algorithm might prefer one side over the other, but only up to some degree. This proves a barrier of Ω(log n) on the competitive ratio for a large class of "natural" algorithms. This class includes all deterministic online algorithms found in the literature so far.Furthermore, we show that our result can be extended to randomized algorithms that locally induce a symmetric distribution over the chosen servers. The Ω(log n)-barrier on the competitive ratio holds for this class of algorithms as well.

Abstract. In the bin covering problem, the goal is to fill as many bins as possible up to a certain minimal level with a given set of items of different sizes. Online variants, in which the items arrive one after another and have to be packed immediately on their arrival without knowledge about the future items, have been studied extensively in the literature. We study the simplest possible online algorithm Dual Next-Fit, which packs all arriving items into the same bin until it is filled and then proceeds with the next bin in the same manner. The competitive ratio of this and any other reasonable online algorithm is 1/2. We study Dual Next-Fit in a probabilistic setting where the item sizes are chosen i.i.d. according to a discrete distribution and we prove that, for every distribution, its expected competitive ratio is at least 1/2 + ǫ for a constant ǫ > 0 independent of the distribution. We also prove an upper bound of 2/3 and better lower bounds for certain restricted classes of distributions. Finally, we prove that the expected competitive ratio equals, for a large class of distributions, the random-order ratio, which is the expected competitive ratio when adversarially chosen items arrive in uniformly random order.

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