Searching in trees, series-parallel and interval orders
SIAM Journal on Computing
On an edge ranking problem of trees and graphs
Discrete Applied Mathematics
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Optimal edge ranking of trees in linear time
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
SIAM Journal on Computing
Protocols for asymmetric communication channels
Journal of Computer and System Sciences
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Improved bounds for asymmetric communication protocols
Information Processing Letters
On an Optimal Split Tree Problem
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
Code and Parse Trees for Lossless Source Encoding
SEQUENCES '97 Proceedings of the Compression and Complexity of Sequences 1997
Searching in random partially ordered sets
Theoretical Computer Science - Latin American theorotical informatics
On the hardness of the minimum height decision tree problem
Discrete Applied Mathematics - Discrete mathematics & data mining (DM & DM)
Lower bounds for asymmetric communication channels and distributed source coding
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Generalization of Binary Search: Searching in Trees and Forest-Like Partial Orders
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Decision trees for entity identification: approximation algorithms and hardness results
Proceedings of the twenty-sixth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Finding an optimal tree searching strategy in linear time
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Edge ranking and searching in partial orders
Discrete Applied Mathematics
An Approximation Algorithm for Binary Searching in Trees
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Approximating Optimal Binary Decision Trees
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Sorting and selection in posets
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Edge ranking of weighted trees
Discrete Applied Mathematics
Minimum average cost testing for partially ordered components
IEEE Transactions on Information Theory
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We study the problem of minimizing the weighted average number of queries to identify an initially unknown object in a poset. We show that for general posets, there cannot be any o(logn)-approximation algorithm unless NP@?TIME(n^O^(^l^o^g^l^o^g^n^)). When the Hasse diagram of the partially ordered set has the structure of a tree, the problem is equivalent to the following tree search problem: in a given rooted tree T=(V,E) a node has been marked and we want to identify it. In order to locate the marked node, we can perform node queries. A node query u asks whether the marked node lies in the subtree rooted at u. A function w:V-Z^+ is given which defines the likelihood for a node to be the one marked, and we want the strategy that minimizes the expected number of queries. Prior to this paper the complexity of this problem had remained open. We prove that the above tree search problem is NP-complete even for the class of trees with diameter at most 4. This results in a complete characterization of the complexity of the problem with respect to the diameter size. In fact, for diameter not larger than 3 we show that the problem is polynomially solvable using a dynamic programming approach. In addition we prove that the problem is NP-complete even for the class of trees of maximum degree at most 16. To the best of our knowledge, the only known result in this direction is that the tree search problem is solvable in O(|V|log|V|) time for trees with degree at most 2 (paths). Our results sharply contrast with those for the variant of the problem where one is interested in minimizing the maximum number of queries. In fact, for the worst case scenario, linear time algorithms are known for finding an optimal search strategy [K. Onak, P. Parys, Generalization of binary search: searching in trees and forest-like partial orders, in: FOCS'06: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Washington, DC, USA, 2006, pp. 379-388; S. Mozes, K. Onak, O. Weimann, Finding an optimal tree searching strategy in linear time, in: SODA'08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008, pp. 1096-1105].