Basic Structures of Some Interconnection Networks

Beginning in the late 1980's attractive alternatives to the n-cubes were proposed as the topologies for larger interconnection networks. These graphs tend to have many vertices as well as good connectivity and routing properties. We will look at recent developments on some newer topologies such as star graphs, alternating group graphs, split-stars, arrangement graphs and generalized (n,k) star graphs. We will present results on routing algorithms, various connectivity measures, structural theorems, augmentation, and open problems. Some applications suggest directed graphs, so the directed versions of above graphs will also be considered.     

Distributed mirror descent method for saddle point problems over directed graphs

In this article, we consider a mini‐max multi‐agent optimization problem where multiple agents cooperatively optimize a sum of local convex–concave functions, each of which is available to one specific agent in a network. To solve the problem, we propose a distributed optimization method by extending classical mirror descent algorithms to the distributed setting. We obtain the convergence of the algorithm under wild conditions that the agent communication follows a directed graph and the related weighted matrices are row stochastic. In particular, when the weighted matrices are restricted to be doubly stochastic, we provide the explicit convergence rate of the algorithm by choosing the stepsize in a suitable way. The proposed algorithm can be viewed as a generalization of the subgradient projection methods since it utilizes a customized Bregman divergence instead of the usual Euclidean squared distance. Finally, some simulation results on a matrix game are presented to illustrate the performance of the algorithm. © 2016 Wiley Periodicals, Inc. Complexity 21: 178–190, 2016     

Average case complexity results for a centering algorithm for linear programming problems under Gaussian distributions

To solve linear programming problems by interior point methods an approximately centered interior point has to be known. Such a point can be found by an algorithmic approach – a so-called phase 1 algorithm or centering algorithm. For random linear programming problems distributed according to the rotation symmetry model, especially with normal distribution, we present probabilistic results on the quality of the origin as starting point and the average number of steps of a centering algorithm.     

Delaunay triangulation and the convex hull ofn points in expected linear time

An algorithm is presented which produces a Delaunay triangulation ofn points in the Euclidean plane in expected linear time. The expected execution time is achieved when the data are (not too far from) uniformly distributed. A modification of the algorithm discussed in the appendix treats most of the non-uniform distributions. The basis of this algorithm is a geographical partitioning of the plane into boxes by the well-known Radix-sort algorithm. This partitioning is also used as a basis for a linear time algorithm for finding the convex hull ofn points in the Euclidean plane.     

Pure adaptive search in global optimization

Pure adaptive seach iteratively constructs a sequence of interior points uniformly distributed within the corresponding sequence of nested improving regions of the feasible space. That is, at any iteration, the next point in the sequence is uniformly distributed over the region of feasible space containing all points that are strictly superior in value to the previous points in the sequence. The complexity of this algorithm is measured by the expected number of iterations required to achieve a given accuracy of solution. We show that for global mathematical programs satisfying the Lipschitz condition, its complexity increases at mostlinearly in the dimension of the problem.This work was supported in part by NATO grant 0119/89.     

二供应商经济批量问题的多项式时间算法

为了从采购费用结构不同的供应商中找到最佳补货策略,考虑一个零售商从两个供应商补货的二供应商经济批量问题.零售商在两个供应商处的采购费用结构分别为复合安装费用和全单位数量折扣费用结构.通过对问题结构性质的分析论证,将问题的可行解转化为一个有向网络,降低问题求解的计算复杂性.综合动态规划和Dijkstra最短路算法证明了该问题是多项式时间可解的.     

On complexity of Minimum Leaf Out-Branching problem

Given a digraph D, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree zero. Gutin, Razgon and Kim [G. Gutin, I. Razgon, E.J. Kim, Minimum leaf out-branching problems, in: Proc. 4th International Conference on Algorithmic Aspects in Information and Management, AAIM’08, in: Lect. Notes Comput. Sci., vol. 5034 2008, pp. 235-246] proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer k, the problem of checking whether there is an out-branching with at most k leaves is polynomial time solvable.     

On strategy improvement algorithms for simple stochastic games

The study of simple stochastic games (SSGs) was initiated by Condon for analyzing the computational power of randomized space-bounded alternating Turing machines. The game is played by two players, MAX and MIN, on a directed multigraph, and when the play terminates at a sink vertex s, MAX wins from MIN a payoff p(s)∈[0,1]. Condon proved that the problem SSG-VALUE—given a SSG, determine whether the expected payoff won by MAX is greater than 1/2 when both players use their optimal strategies—is in NP∩coNP. However, the exact complexity of this problem remains open, as it is not known whether the problem is in P or is hard for some natural complexity class. In this paper, we study the computational complexity of a strategy improvement algorithm by Hoffman and Karp for this problem. The Hoffman–Karp algorithm converges to optimal strategies of a given SSG, but no non-trivial bounds were previously known on its running time. We prove a bound of O(n²/n) on the convergence time of the Hoffman–Karp algorithm, and a bound of O(²0.78n) on a randomized variant. These are the first non-trivial upper bounds on the convergence time of these strategy improvement algorithms.     

基于固定序的Bellman-Ford算法的改进

固定序算法是Bellman-Ford算法的一种基本改进算法。为了改变固定序算法在稀疏图上的劣势,本文通过预先订制参与迭代的点的计算顺序,对该算法进行了改进。实验表明,在稀疏图上, 改进后的算法相对于原算法计算效率提高了近50%, 并能够与国际流行的先进先出算法相媲美。本文的工作表明,固定序算法不仅在大规模稠密图上具有明显的优势,而且在稀疏图上也具有很强的竞争力。     

Improving Hit-and-Run for global optimization

Improving Hit-and-Run is a random search algorithm for global optimization that at each iteration generates a candidate point for improvement that is uniformly distributed along a randomly chosen direction within the feasible region. The candidate point is accepted as the next iterate if it offers an improvement over the current iterate. We show that for positive definite quadratic programs, the expected number of function evaluations needed to arbitrarily well approximate the optimal solution is at most O(n^5/2) wheren is the dimension of the problem. Improving Hit-and-Run when applied to global optimization problems can therefore be expected to converge polynomially fast as it approaches the global optimum.Paper presented at the II. IIASA-Workshop on Global Optimization, December 9–14, 1990, Sopron (Hungary).     

A dual ascent approach for steiner tree problems on a directed graph

The Steiner tree problem on a directed graph (STDG) is to find a directed subtree that connects a root node to every node in a designated node setV. We give a dual ascent procedure for obtaining lower bounds to the optimal solution value. The ascent information is also used in a heuristic procedure for obtaining feasible solutions to the STDG. Computational results indicate that the two procedures are very effective in solving a class of STDG's containing up to 60 nodes and 240 directed/120 undirected arcs. The directed spanning tree and uncapacitated plant location problems are special cases of the STDG. Using these relationships, we show that our ascent procedure can be viewed as a generalization ofboth the Chu-Liu-Edmonds directed spanning tree algorithm and the Bilde-Krarup-Erlenkotter ascent algorithm for the plant location problem. The former comparison yields a dual ascent interpretation of the steps of the directed spanning tree algorithm.     

Distributions of Runs and Consecutive Systems on Directed Trees

In this paper we study exact distributions of runs on directed trees. On the assumption that the collection of random variables indexed by the vertices of a directed tree has a directed Markov distribution, the exact distribution theory of runs is extended from based on random sequences to based on directed trees. The distribution of the number of success runs of a specified length on a directed tree along the direction is derived. A consecutive-k-out-of-n:F system on a directed tree is introduced and investigated. By assuming that the lifetimes of the components are independent and identically distributed, we give the exact distribution of the lifetime of the consecutive system. The results are not only theoretical but also suitable for computation.     

Probabilistic analysis of an approximation algorithm for the traveling salesman problem on unbounded above instances

The probabilistic analysis of a modification of the approximation algorithm “Nearest City” is presented for the traveling salesman minimization problem. The case is considered when the entries of the distance matrix are some independent identically distributed random variables with the values in the range [a _n ,∞), a _n > 0, that are unbounded above and have a truncated normal or exponential distribution. We obtain the estimates of relative errors, the failure probabilities, and also the conditions of asymptotic optimality of the algorithm. The modification of the algorithm allows us to perform analysis so that the obtained results are true for the traveling salesman problems in the cases of both directed and undirected graphs.     

On the complexity of the exchange algorithm for minimax optimization problems

We present an exchange algorithm for the solution of minimax optimization problems involving convex functions. For a certain class of functions, the complexity of this algorithm is shown to be either linear in the number of functions, or at least squared in that number.     

有向网络中具有一个枢纽点的最小支撑树的计算方法

对有向网络中具有一个枢纽点的支撑树的问题和性质进行了研究,给出了在有向网络图中寻找以某一定点为枢纽点的最小支撑树的计算方法,并对算法的复杂性进行了讨论,最后将该算法应用于实际算例的计算．     

A multicast routing algorithm based on searching in directed graph

Aiming at constructing a delay and delay variation bounded Steiner tree in the real-time streaming media communication, in this paper, we discuss a multicast routing algorithm based on searching a directed graph (MRASDH). During the process of the construction of the multicast tree, some nodes and links in the network topology do not affect the outcome of the constructed tree. Therefore, based on the thought of shrinking the search space through deleting these non-relative nodes and edges to the utmost, the ant algorithm is utilized to generate a directed sub-graph of the network topology for each destination node, in which each node owns a bounded out-degree. And all these sub-graphs can be merged into a new directed graph that serves as the new search space. In the new space, the simulated annealing algorithm is applied to obtain a multicast tree that satisfies the condition for the optimization. The performance analysis and simulation results demonstrate that this algorithm can effectively construct a delay and delay variation bounded multicast tree. They also show that the algorithm have lower time complexity than the current ones, which means a much better result would be achieved when the system scale rises greatly.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

A recognition algorithm for the intersection graphs of directed paths in directed trees

Consider a finite family of non-empty sets. The intersection graph of this family is obtained by representing each set by a vertex, two vertices being connected by an edge if and only if the corresponding sets intersect. The intersection graph of a family of directed paths in a directed tree is called a directed path graph. In this paper we present an efficient algorithm which constructs to a given graph a representation by a family of directed paths on a directed tree, if one exists. Also, we prove that a graph is a proper directed path graph if and only if it is a directed path graph.     

基于自适应参数校正策略求解SDP的Mehrotra型内点算法

最近,Salahi对线性规划提出了一个基于新的自适应参数校正策略的Mehrotra型预估-校正算法,该策略使其在不使用安全策略的情况下,证明了算法的多项式迭代复杂界.本文将这一算法推广到半定规划的情形.通过利用Zhang的对称化技术,得到了算法的多项式迭代复杂界,这与求解线性规划的相应算法有相同的迭代复杂性阶.