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Approximation Strategies for Generalized Binary Search in Weighted Trees

Authors Dariusz Dereniowski, Adrian Kosowski, Przemyslaw Uznanski, Mengchuan Zou

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  • Approximation Algorithm
  • Adaptive Algorithm
  • Graph Search
  • Binary Search
  • Vertex Ranking
  • Trees

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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\{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 (QPTAS): for any 0 < epsilon < 1, there exists a (1+epsilon)-approximation strategy with a computation time of n^O(log n / epsilon^2). Thus, the problem is not APX-hard, unless NP is contained in 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 tilde-O(log n)-approximation approaches, where the tilde-O-notation disregards O(poly log log n)-factors.

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Dariusz Dereniowski, Adrian Kosowski, Przemyslaw Uznanski, and Mengchuan Zou. Approximation Strategies for Generalized Binary Search in Weighted Trees. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 84:1-84:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ICALP.2017.84

Author Details

Dariusz Dereniowski
Adrian Kosowski
Przemyslaw Uznanski
Mengchuan Zou

References

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