Fixed-Parameter Algorithms for CLOSEST STRING and Related Problems

CLOSEST STRING is one of the core problems in the field of consensus word analysis with particular importance for computational biology. Given k strings of the same length and a nonnegative integer d , find a ``center string' s such that none of the given strings has the Hamming distance greater than d from s . CLOSEST STRING is NP-complete. In biological applications, however, d is usually very small. We show how to solve CLOSEST STRING in linear time for fixed d —the exponential growth in d is bounded by O(d ^d ) . We extend this result to the closely related problems d -MISMATCH and DISTINGUISHING STRING SELECTION. Moreover, we also show that CLOSEST STRING is solvable in linear time when k is fixed and d is arbitrary. In summary, this means that CLOSEST STRING is fixed-parameter tractable with respect to parameter d and with respect to parameter k . Finally, the practical usefulness of our findings is substantiated by some experimental results.     

Parallel Algorithms for Maximum Matching in Complements of Interval Graphs and Related Problems

Given a set of n intervals representing an interval graph, the problem of finding a maximum matching between pairs of disjoint (nonintersecting) intervals has been considered in the sequential model. In this paper we present parallel algorithms for computing maximum cardinality matchings among pairs of disjoint intervals in interval graphs in the EREW PRAM and hypercube models. For the general case of the problem, our algorithms compute a maximum matching in O( log ³ n) time using O(n/ log ² n) processors on the EREW PRAM and using n processors on the hypercubes. For the case of proper interval graphs, our algorithm runs in O( log n ) time using O(n) processors if the input intervals are not given already sorted and using O(n/ log n ) processors otherwise, on the EREW PRAM. On n -processor hypercubes, our algorithm for the proper interval case takes O( log n log log n ) time for unsorted input and O( log n ) time for sorted input. Our parallel results also lead to optimal sequential algorithms for computing maximum matchings among disjoint intervals. In addition, we present an improved parallel algorithm for maximum matching between overlapping intervals in proper interval graphs. Received November 20, 1995; revised September 3, 1998.     

任意图支配集精确算法回顾

该文综述了任意图支配集精确算法分析和设计的新进展.支配集问题是经典NP完全问题,很多问题都能与它相联系.我们针对最小支配集、最大独立集、最小独立支配集、最小连通支配集、最小加权支配集问题提供了详尽算法描述和实例说明,以使文章自包含方便阅读.文中还讨论了诸如分支简化策略、复杂度分析、测度分析、记忆等技术.自Claude Berge首次准确阐述现代图支配概念后,经过很长一段时期的沉寂,关于指数时间精确算法设计的研究热情在过去五年中显著增涨.除回顾这些最新成果之外,作者还盼望国内研究团体能更加重视这个快速发展的研究领域.     

Parallel Algorithms for Series Parallel Graphs and Graphs with Treewidth Two 1

In this paper a parallel algorithm is given that, given a graph G=(V,E) , decides whether G is a series parallel graph, and, if so, builds a decomposition tree for G of series and parallel composition rules. The algorithm uses O(log \kern -1pt |E|log ^\ast \kern -1pt |E|) time and O(|E|) operations on an EREW PRAM, and O(log \kern -1pt |E|) time and O(|E|) operations on a CRCW PRAM. The results hold for undirected as well as for directed graphs. Algorithms with the same resource bounds are described for the recognition of graphs of treewidth two, and for constructing tree decompositions of treewidth two. Hence efficient parallel algorithms can be found for a large number of graph problems on series parallel graphs and graphs with treewidth two. These include many well-known problems like all problems that can be stated in monadic second-order logic. Received July 15, 1997; revised January 29, 1999, and June 23, 1999.