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Approximation Strategies for Generalized Binary Search in Weighted Trees

Abstract : We consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node $t$ in a given tree $T$. Upon querying a node $v$ of the tree, the strategy receives as a reply an indication of the connected component of $T\setminus\{v\}$ containing the target $t$. The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total query cost for a worst-case choice of the target. Designing an optimal strategy for a weighted tree search instance is known to be strongly NP-hard, in contrast to the unweighted variant of the problem which can be solved optimally in linear time. Here, we show that weighted tree search admits a quasi-polynomial time approximation scheme: for any $0 < \varepsilon < 1$, there exists a $(1+\varepsilon)$-approximation strategy with a computation time of $n^{O(\log n / \varepsilon^2)}$. Thus, the problem is not APX-hard, unless $NP \subseteq DTIME(n^{O(\log n)})$. By applying a generic reduction, we obtain as a corollary that the studied problem admits a polynomial-time $O(\sqrt{\log n})$-approximation. This improves previous $\hat O(\log n)$-approximation approaches, where the $\hat O$-notation disregards $O(\mathrm{poly}\log\log n)$-factors.
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Contributor : Adrian Kosowski Connect in order to contact the contributor
Submitted on : Sunday, February 26, 2017 - 8:21:53 PM
Last modification on : Friday, January 21, 2022 - 3:19:23 AM
Long-term archiving on: : Saturday, May 27, 2017 - 12:23:42 PM


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Dariusz Dereniowski, Adrian Kosowski, Przemyslaw Uznanski, Mengchuan Zou. Approximation Strategies for Generalized Binary Search in Weighted Trees . ICALP 2017 - 44th International Colloquium on Automata, Languages, and Programming, 2017, Warsaw, Poland. pp.84:1--84:14, ⟨10.4230/LIPIcs.ICALP.2017.84⟩. ⟨hal-01470935⟩



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