We study the problem of evaluating a discrete function by adaptively querying the values of its variables until the values read uniquely determine the value of the function. Reading the value of a variable is done at the expense of some cost, and the goal is to design a strategy (decision tree) for evaluating the function incurring as little cost as possible in the worst case or in expectation (according to a prior distribution on the possible variables assignments). Except for particular cases of the problem, in general, only the minimization of one of these two measures is addressed in the literature. However, there are instances of the problem for which the minimization of one measure leads to a strategy with a high cost with respect to the other measure (even exponentially bigger than the optimal). We provide a new construction which can guarantee a trade-off between the two criteria. More precisely, given a decision tree guaranteeing expected cost @math and a decision tree guaranteeing worst cost @math our method can guarantee for any chosen trade-off value @math to produce a decision tree whose worst cost is @math and whose expected cost is @math These bounds are improved for the relevant case of uniform testing costs. Motivated by applications, we also study a variant of the problem where the cost of reading a variable depends on the variable's value. We provide an @math approximation algorithm for the minimization of the worst cost measure, which is best possible under the assumption @math .
A number of algorithms with different time complexities were proposed to construct decision trees with minimum expected path length (expected testing cost in DFEP terminology) among the decision trees with depth (worst testing cost) at most @math , where @math is a given integer @cite_1 @cite_18 @cite_15 .
@cite_18 @cite_1 @cite_15
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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-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. We give a (ln n+ 1)-approximation for the decision tree (DT) problem. An instance of DT is a set of mbinary tests T= (T 1 , ..., T m ) and a set of nitems X= (X 1 , ..., X n ). The goal is to output a binary tree where each internal node is a test, each leaf is an item and the total external path length of the tree is minimized. Total external path length is the sum of the depths of all the leaves in the tree. DT has a long history in computer science with applications ranging from medical diagnosis to experiment design. It also generalizes the problem of finding optimal average-case search strategies in partially ordered sets which includes several alphabetic tree problems. Our work decreases the previous upper bound on the approximation ratio by a constant factor. We provide a new analysis of the greedy algorithm that uses a simple accounting scheme to spread the cost of a tree among pairs of items split at a particular node. We conclude by showing that our upper bound also holds for the DT problem with weighted tests. There are many classical problems in P whose time complexities have not been improved over the past decades. Recent studies of "Hardness in P" have revealed that, for several of such problems, the current fastest algorithm is the best possible under some complexity assumptions. To bypass this difficulty, (SODA 2017) introduced the concept of fully polynomial FPT algorithms. For a problem with the current best time complexity @math , the goal is to design an algorithm running in @math time for a parameter @math and a constant @math . In this paper, we investigate the complexity of graph problems in P parameterized by tree-depth, a graph parameter related to tree-width. We show that a simple divide-and-conquer method can solve many graph problems, including Weighted Matching, Negative Cycle Detection, Minimum Weight Cycle, Replacement Paths, and 2-hop Cover, in @math time or @math time, where @math is the tree-depth of the input graph. Because any graph of tree-width @math has tree-depth at most @math , our algorithms also run in @math time or @math time. These results match or improve the previous best algorithms parameterized by tree-width. Especially, we solve an open problem of fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width posed by In this paper the minimum spanning tree problem with uncertain edge costs is discussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within O(log^1^-^@en) for any @e>0 unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within O(logn) unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance bound of O(log^2n) for min-max and 2-stage min-max problems are also proposed.