Abstract
Local computation algorithms (LCAs) produce small parts of a single solution to a given search problem using time and space sublinear in the size of the input. In this work we present LCAs whose time complexity (and usually also space complexity) is independent of the input size. Specifically, we give (1) a \((1-\epsilon )\)-approximation LCA to the maximal weighted base of a graphic matroid (i.e., maximal acyclic edge set), (2) LCAs for approximating multicut and integer multicommodity flow on trees, and (3) a local reduction of weighted matching to any unweighted matching LCA, such that the running time of the weighted matching LCA is also independent of the edge weight function.
Y. Mansour—Supported in part by a grant from the Israel Science Foundation, by a grant from United States-Israel Binational Science Foundation (BSF), by a grant from the Israeli Ministry of Science (MoS) and the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11).
B. Patt-Shamir—Supported in part by the Israel Science Foundation (grant No. 1444/14) and by the Israel Ministry of Science and Technology.
S. Vardi—Supported in part by the Google Europe Fellowship in Game Theory.
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- 1.
We assume the standard uniform-cost RAM model, in which the word size is \(O(\log {n})\) bits, where n is the input size.
- 2.
The failure probability of an LCA A (cf. [7]) is the probability, taken over coin flips of A, that the running time of A for any query exceeds its stated running time (and not the probability that A errs).
- 3.
We typically assume that vertex degrees are bounded by a constant.
- 4.
We note that while our algorithm for MWM runs in constant time, independently of the size of the graph and of the edge weights, its approximation guarantee is much worse than that of [2], whose approximation factor is \((1-\epsilon )\).
- 5.
In case of a randomized algorithm, expectation is over its random choices.
- 6.
Note that the LCA is not allowed to deviate from the enduring memory bound.
- 7.
The ratio is \(\frac{1}{4}\) when all capacities are even, and it tends to \(\frac{1}{4}\) as \(c_{\min }\rightarrow \infty \). For \(c_{\min }=1\) the approximation ratio is 0.
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The authors would like to thank the anonymous reviewers for their useful feedback.
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Mansour, Y., Patt-Shamir, B., Vardi, S. (2015). Constant-Time Local Computation Algorithms. In: Sanità, L., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2015. Lecture Notes in Computer Science(), vol 9499. Springer, Cham. https://doi.org/10.1007/978-3-319-28684-6_10
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