Tree spanners on chordal graphs: complexity and algorithms

A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The T t-S problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil (SIAM J. Discrete Math. 8 (1995) 359–387) by showing that, for any t4, T t-S is NP-complete even on chordal graphs of diameter at most t+1 (if t is even), respectively, at most t+2 (if t is odd). Then we point out that every chordal graph of diameter at most t−1 (respectively, t−2) admits a tree t-spanner whenever t2 is even (respectively, t3 is odd), and such a tree spanner can be constructed in linear time.

The complexity status of T 3-S still remains open for chordal graphs, even on the subclass of undirected path graphs that are strongly chordal as well. For other important subclasses of chordal graphs, such as very strongly chordal graphs (containing all interval graphs), 1-split graphs (containing all split graphs) and chordal graphs of diameter at most 2, we are able to decide T 3-S efficiently.     

有不同中断时间代价的一致并行抢先调度问题

提出了具有不同中断时间代价的抢先调度问题(P|ptmn(δi)|Cmax)．该问题在工程任务分配、分布式计算和网络通信等实际问题中有着广泛的应用背景．首先证明了这个问题是一个NP难优化问题．并给出了一个时间复杂度为O(nlogn)的近似算法,其近似度为5／3．算法的特点是结合中断时间δi来应用LPT思想,而不只是把它应用到任务i的执行时间pi上,从而避免了LPT算法在最坏情形下的近似度差的问题．在算法的关键部分,运用了均分的技巧来提高任务执行的并行性,进一步提高了近似度．     

Flows in dynamic networks with aggregate arc capacities

Dynamic networks are characterized by transit times on edges. Dynamic flow problems consider transshipment problems in dynamic networks. We introduce a new version of dynamic flow problems, called bridge problem. The bridge problem has practical importance and raises interesting theoretical issues. We show that the bridge problem is NP-complete. Traditional static flow techniques for solving dynamic flow problems do not extend to the new problem. We give a linear programming formulation for the bridge problem which is based on the time-expanded network of the original dynamic network.     

Nash Stability in Additively Separable Hedonic Games and Community Structures

We prove that the problem of deciding whether a Nash stable partition exists in an Additively Separable Hedonic Game is NP-complete. We also show that the problem of deciding whether a non trivial Nash stable partition exists in an Additively Separable Hedonic Game with non-negative and symmetric preferences is NP-complete. We motivate our study of the computational complexity by linking Nash stable partitions in Additively Separable Hedonic Games to community structures in networks. Our results formally justify that computing community structures in general is hard. The research is partly sponsored by the company Cofman ().     

一种分布式系统中最优复制的策略分析

在网络中定位最优复制以最小化通讯代价。假定网络采用read-one-write-all策略来保证网络数据一致性,那么存在一个决定复制定位的最优化问题。提出了研究复制问题中读、写比率以确定最优化通讯代价。问题可转换成一个0- 1线性规划问题,并将此问题扩展为一个P中值问题,可以证明这个问题是NP-complete的问题,并提出了一种多项式时间内的此问题求解算法。     

一种新的WDM光网络波长分配算法

分析比较了目前WDM光网络中提出的各种固定路由选路下的波长分配算法,提出了一种新的固定路由选路的波长分配算法,并在环网、Mesh网和类教育网中,对新算法和已有算法进行性能仿真,仿真结果表明,新算法减小了网络的阻塞概率,性能优于已有的算法。     

Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard

Let G=(V,E) be an undirected graph and C a subset of vertices. If the sets B_r(v)∩C, vV (respectively, vVC), are all nonempty and different, where B_r(v) denotes the set of all points within distance r from v, we call C an r-identifying code (respectively, an r-locating-dominating code). We prove that, given a graph G and an integer k, the decision problem of the existence of an r-identifying code, or of an r-locating-dominating code, of size at most k in G, is NP-complete for any r.     

Complexities of some interesting problems on spanning trees

This paper deals with the complexity issues of some new interesting spanning tree problems. Here we define some new spanning tree problems by imposing various constraints and restrictions on graph parameters and present relevant results. Also we introduce a new notion of “set version” of some decision problems having integer K<|V| as a parameter in the input instance, where we replace K by a set X⊆|V|. For example, the set version of Maximum Leaf Spanning Tree problem asks whether there exists a spanning tree in G that contains X as a subset of the leaf set. We raise the issue of whether the set versions of NP-complete problems are as hard as the original problems and prove that although in some cases the set versions are easier to solve, this is not necessarily true in general.     

The automation of syllogistic

In the first paper of this series it was shown that any unquantified formula p in the collection MLSSF (multilevel syllogistic extended with the singleton operator and the predicate Finite) can be decomposed as a disjunction of set-theoretic formulae called syllogistic schemes. The syllogistic schemes are satisfiable and no two of them have a model in common, therefore the previous result already implied the decidability of the class MLSSF by simply checking if the set of syllogistic schemes associated with the given formula is empty.In the first section of this paper a new and improved searching algorithm for syllogistic schemes is introduced, based on a proof of existence of a minimum effort scheme for any given satisfiable formula in MLSF. The algorithm addressed above can be piloted quite effectively even though it involves backtracking.In the second part of the paper, complexity issues are studied by showing that the class of () _o ¹ -simple prenex formulae (an extension of MLS) has a decision problem which is NP-complete. The decision algorithm that proves the membership of this decision problem to NP can be seen as a different decision algorithm for MLS.Research supported by ENI and ENIDATA within the AXL project.     

High parallelism and a proof procedure I: Theoretical considerations

This paper presents the theoretical basis of a proof procedure, which allows a high degree of parallel processing. The theoretical method is based upon the works of Prawitz's improved proof procedure and Robinson's unification algorithm. The input to the method is a set of clauses (or alternatively, well-formed formulae). The output of the method is the solution to the problem, if it exists. To overcome the inefficiency of the theoretical approach, we outline the main steps of a practical proof procedure. Besides parallel processing, the proof procedure works with bit manipulation rather than symbol manipulation as found in most of the existing proof mechanisms.