In many fault-detection problems, we want to identify defective items from a set of n items using the minimum number of tests. Group testing is a scenario in which each test is on a subset of items and determines whether the subset contains at least one defective item. In practice, the number d of defective items is often unknown in advance. In this paper, we present a new algorithm for the above group testing problem and prove that it has very good performance guarantee. More specifically, the number of tests used by the new algorithm is bounded from above by d log(n/d) + 3d + O(log2 d). The new algorithm is designed based on a zig-zag approach that has not been studied before and is intuitive and easy to implement. When 0 < d < ρ0n where ρ0 = 1 − 4/e2 = 0.45…, which holds for most practical applications, our new algorithm has better performance guarantee than any previous best result. Computational results show that the new algorithm has very good practical performances.

The firefighter problem is the following discrete-time game on a graph. Initially, a fire starts at a vertex of the graph. In each round, a firefighter protects one vertex not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate of a graph is the average percentage of vertices that can be saved when a fire starts randomly at one vertex of the graph, which measures the defense ability of a graph as a whole. In this paper, we study the surviving rates of graphs with bounded treewidth. We prove that the surviving rate of every n-vertex outerplanar graph is at least $1-\Theta(\frac{\log n}{n})$, which is asymptotically tight. We also prove that if k firefighters are available in each round, then the surviving rate of an n-vertex graph with treewidth at most k is at least $1-O(\frac{k^{2}\log n}{n})$. Furthermore, we show that the greedy strategy of Hartnell and Li [Congr. Numer., 145 (2000), pp. 187–192] for trees saves at least $1-\Theta(\frac{\log n}{n})$ percent of vertices on average for an n-vertex tree. Our results settle a conjecture and two problems of Cai and Wang [SIAM J. Discrete Math., 23 (2009), pp. 1814–1826] in affirmative.

This paper describes a transformational method applied to the core component of role-based access control (RBAC), to derive efficient implementations from a specification based on the ANSI standard for RBAC. The method is based on the idea of incrementally maintaining the result of expensive set operations, where a new method is described and used for systematically deriving incrementalization rules. We calculate precise complexities for three variants of efficient implementations as well as for a straightforward implementation based on the specification. We describe successful prototypes and experiments for the efficient implementations and for automatically generating efficient implementations from straightforward implementations.