We study packing LPs in an online model where the columns are presented to the algorithm in random order. This natural problem was investigated in various recent studies motivated, e.g., by online ad allocations and yield management where rows correspond to resources and columns to requests specifying demands for resources. Our main contribution is a 1 − O( (log d) /B)-competitive online algorithm, where d denotes the column sparsity, i.e., the maximum number of resources that occur in a single column, and B denotes the capacity ratio B, i.e., the ratio between the capacity of a resource and the maximum demand for this resource. In other words, we achieve a (1−ǫ)-approximation if the capacity ratio satisfies B = Ω( log d ǫ 2 ), which is known to be best-possible for any (randomized) online algorithms.Our result improves exponentially on previous work with respect to the capacity ratio. In contrast to existing results on packing LP problems, our algorithm does not use dual prices to guide the allocation of resources over time. Instead, the algorithm simply solves, for each request, a scaled version of the partially known primal program and randomly rounds the obtained fractional solution to obtain an integral allocation for this request. We show that this simple algorithmic technique is not restricted to packing LPs with large capacity ratio of order Ω(log d), but it also yields close-to-optimal competitive ratios if the capacity ratio is bounded by a constant. In particular, we prove an upper bound on the competitive ratio of Ω(d − 1 /(B−1) ), for any B ≥ 2. In addition, we show that our approach can be combined with VCG payments and obtain an incentive compatible (1 − ǫ)-competitive mechanism for packing LPs with B = Ω( log m ǫ 2 ), where m is the number of constraints. Finally, we apply our technique to the generalized assignment problem for which we obtain the first online algorithm with competitive ratio O(1). * Supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD). † Supported by the Studienstiftung des deutschen Volkes. ‡ Supported by the DFG GRK/1298 "AlgoSyn". 1 We follow the convention of using log k to denote log k = max{log 2 k, 1} in asymptotic statements.
We consider the (block-angular) min-max resource sharing problem, which is defined as follows. Given finite sets R of resources and C of customers, a convex set B c , called block, and a convex function g c :As usual we assume that g c can be computed efficiently and we have a constant σ ≥ 1 and oracle functionsWe describe a simple algorithm which solves this problem with an approximation guarantee σ (1 + ω) for any ω > 0, and whose running time is O(θ (|C|+|R|) log |R|(log log |R|+ω −2 )) for any fixed σ ≥ 1, where θ is the time for an oracle call. This generalizes and improves various previous results. We also prove other bounds and describe several speed-up techniques. In particular, we show how to parallelize the algorithm efficiently. In addition we review another algorithm, variants of which were studied before. We show that this algorithm is almost as fast in theory, but it was not competitive in our experiments. Our work was motivated mainly by global routing in chip design. Here the blocks are mixedinteger sets (whose elements are associated with Steiner trees), and we combine our algorithm with randomized rounding. We present experimental results on instances 123 2 D. Müller et al.resulting from recent industrial chips, with millions of customers and resources. Our algorithm solves these instances nearly optimally in less than two hours.Keywords Min-max resource sharing · Fractional packing · Fully polynomial approximation scheme · Parallelization · Global routing · Chip design Mathematics Subject Classification (2010) 90C27 · 90C90 · 90C59 · 90C48 · 90C06
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