7 research outputs found
Deterministic and Probabilistic Binary Search in Graphs
We consider the following natural generalization of Binary Search: in a given
undirected, positively weighted graph, one vertex is a target. The algorithm's
task is to identify the target by adaptively querying vertices. In response to
querying a node , the algorithm learns either that is the target, or is
given an edge out of that lies on a shortest path from to the target.
We study this problem in a general noisy model in which each query
independently receives a correct answer with probability (a
known constant), and an (adversarial) incorrect one with probability .
Our main positive result is that when (i.e., all answers are
correct), queries are always sufficient. For general , we give an
(almost information-theoretically optimal) algorithm that uses, in expectation,
no more than queries, and identifies the target correctly with probability at
leas . Here, denotes the
entropy. The first bound is achieved by the algorithm that iteratively queries
a 1-median of the nodes not ruled out yet; the second bound by careful repeated
invocations of a multiplicative weights algorithm.
Even for , we show several hardness results for the problem of
determining whether a target can be found using queries. Our upper bound of
implies a quasipolynomial-time algorithm for undirected connected
graphs; we show that this is best-possible under the Strong Exponential Time
Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs
with non-uniform node querying costs, the problem is PSPACE-complete. For a
semi-adaptive version, in which one may query nodes each in rounds, we
show membership in in the polynomial hierarchy, and hardness
for
Approximation Strategies for Generalized Binary Search in Weighted Trees
International audienceWe consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node in a given tree . Upon querying a node of the tree, the strategy receives as a reply an indication of the connected component of containing the target . 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 , there exists a -approximation strategy with a computation time of . Thus, the problem is not APX-hard, unless . By applying a generic reduction, we obtain as a corollary that the studied problem admits a polynomial-time -approximation. This improves previous -approximation approaches, where the -notation disregards -factors
Binary Identification Problems for Weighted Trees
The Binary Identification Problem for weighted trees asks for the minimum cost strategy (decision tree) for identifying a node in an edge weighted tree via testing edges. Each edge has assigned a different cost, to be paid for testing it. Testing an edge e reveals in which component of T â e lies the vertex to be identified. We give a complete characterization of the computational complexity of this problem with respect to both tree diameter and degree. In particular, we show that it is strongly NP-hard to compute a minimum cost decision tree for weighted trees of diameter at least 6, and for trees having degree three or more. For trees of diameter five or less, we give a polynomial time algorithm. More- over, for the degree 2 case, we significantly improve the straightforward O(n^3) dynamic programming approach, and provide an O(n^2) time algorithm. Finally, this work contains the first approximate decision tree construction algorithm that breaks the barrier of factor logn
The Binary Identification Problems for Weighted Trees
The Binary Identification Problem for weighted trees asks for the minimum cost strategy (decision tree) for identifying a vertex in an edge weighted tree via testing edges. Each edge has assigned a different cost, to be paid for testing it. Testing an edge e reveals in which component of T 12 e lies the vertex to be identified. We give a complete characterization of the computational complexity of this problem with respect to both tree diameter and degree. In particular, we show that it is strongly NP-hard to compute a minimum cost decision tree for weighted trees of diameter at least 6, and for trees having degree three or more. For trees of diameter five or less, we give a polynomial time algorithm. Moreover, for the degree 2 case, we significantly improve the straightforward O(n3 ) dynamic programming approach, and provide an O(n2 ) time algorithm. Finally, this work contains the first approximate decision tree construction algorithm that breaks the barrier of factor log n
The Binary Identification Problems for Weighted Trees
The Binary Identification Problem for weighted trees asks for the minimum cost strategy (decision tree) for identifying a vertex in an edge weighted tree via testing edges. Each edge has assigned a different cost, to be paid for testing it. Testing an edge e reveals in which component of T â e lies the vertex to be identified. We give a complete characterization of the computational complexity of this problem with respect to both tree diameter and degree. In particular, we show that it is strongly NP-hard to compute a minimum cost decision tree for weighted trees of diameter at least 6, and for trees having degree three or more. For trees of diameter five or less, we give a polynomial time algorithm. Moreover, for the degree 2 case, we significantly improve the straightforward O(n^3) dynamic programming approach, and provide an O(n^2) time algorithm. Finally, this work contains the first approximate decision tree construction algorithm that breaks the barrier of factor logn