A Cut-Based Algorithm for the Nonlinear Dual of the Minimum Cost Network Flow Problem

We consider a convex, or nonlinear, separable minimization problem with constraints that are dual to the minimum cost network flow problem. We show how to reduce this problem to a polynomial number of minimum s,t-cut problems. The solution of the reduced problem utilizes the technique for solving integer programs on monotone inequalities in three variables, and a so-called proximity-scaling technique that reduces a convex problem to its linear objective counterpart. The problem is solved in this case in a logarithmic number of calls, O(log U), to a minimum cut procedure, where U is the range of the variables. For a convex problem on n variables the minimum cut is solved on a graph with O(n²) nodes. Among the consequences of this result is a new cut-based scaling algorithm for the minimum cost network flow problem. When the objective function is an arbitrary nonlinear function we demonstrate that this constrained problem is solved in pseudopolynomial time by applying a minimum cut procedure to a graph on O(nU) nodes.     

Closed-Form Solution of Visual-Inertial Structure from Motion

This paper investigates the visual-inertial structure from motion problem. A simple closed form solution to this problem is introduced. Special attention is devoted to identify the conditions under which the problem has a finite number of solutions. Specifically, it is shown that the problem can have a unique solution, two distinct solutions and infinite solutions depending on the trajectory, on the number of point-features and on their layout and on the number of camera images. The investigation is also performed in the case when the inertial data are biased, showing that, in this latter case, more images and more restrictive conditions on the trajectory are required for the problem resolvability.     

Parallel solution of the subset‐sum problem: an empirical study

The subset‐sum problem is a well‐known NP‐complete combinatorial problem that is solvable in pseudo‐polynomial time, that is, time proportional to the number of input objects multiplied by the sum of their sizes. This product defines the size of the dynamic programming table used to solve the problem. We show how this problem can be parallelized on three contemporary architectures, that is, a 128‐processor Cray Extreme Multithreading (XMT) massively multithreaded machine, a 16‐processor IBM x3755 shared memory machine, and a 240‐core NVIDIA FX 5800 graphics processing unit (GPU). We show that it is straightforward to parallelize this algorithm on the Cray XMT primarily because of the word‐level locking that is available on this architecture. For the other two machines, we present an alternating word algorithm that can implement an efficient solution. Our results show that the GPU performs well for problems whose tables fit within the device memory. Because GPUs typically have memories in the order of 10GB, such architectures are best for small problem sizes that have tables of size approximately 10¹⁰. The IBM x3755 performs very well on medium‐sized problems that fit within its 64‐GB memory but has poor scalability as the number of processors increases and is unable to sustain performance as the problem size increases. This machine tends to saturate for problem sizes of 10¹¹ bits. The Cray XMT shows very good scaling for large problems and demonstrates sustained performance as the problem size increases. However, this machine has poor scaling for small problem sizes; it performs best for problem sizes of 10¹² bits or more. The results in this paper illustrate that the subset‐sum problem can be parallelized well on all three architectures, albeit for different ranges of problem sizes. The performance of these three machines under varying problem sizes show the strengths and weaknesses of the three architectures. Copyright © 2012 John Wiley & Sons, Ltd.     

Fast Computation of Optimal Contact Forces

We consider the problem of computing the smallest contact forces, with point-contact friction model, that can hold an object in equilibrium against a known external applied force and torque. It is known that the force optimization problem (FOP) can be formulated as a semidefinite programming problem (SDP) or a second-order cone problem (SOCP), and thus, can be solved using several standard algorithms for these problem classes. In this paper, we describe a custom interior-point algorithm for solving the FOP that exploits the specific structure of the problem, and is much faster than these standard methods. Our method has a complexity that is linear in the number of contact forces, whereas methods based on generic SDP or SOCP algorithms have complexity that is cubic in the number of forces. Our method is also much faster for smaller problems. We derive a compact dual problem for the FOP, which allows us to rapidly compute lower bounds on the minimum contact force and certify the infeasibility of a FOP. We use this dual problem to terminate our optimization method with a guaranteed accuracy. Finally, we consider the problem of solving a family of FOPs that are related. This occurs, for example, in determining whether force closure occurs, in analyzing the worst case contact force required over a set of external forces and torques, and in the problem of choosing contact points on an object so as to minimize the required contact force. Using dual bounds, and a warm-start version of our FOP method, we show how such families of FOPs can be solved very efficiently.     

On the k-path partition of graphs

The k-path partition problem is to partition a graph into the minimum number of paths, so that none of them has length more than k, for a given positive integer k. The problem is a generalization of the Hamiltonian path problem and the problem of partitioning a graph into the minimum number of paths. The k-path partition problem remains NP-complete on the class of chordal bipartite graphs if k is part of the input, and we show that it is NP-complete on the class of comparability graphs even for k=3. On the positive side, we present a polynomial-time solution for the problem, with any k, on bipartite permutation graphs, which form a subclass of chordal bipartite graphs.     

An approach to estimating the average-case complexity of postoptimality analysis of discrete optimization problems

It is shown that an algorithm polynomial on average with respect to μ and that determines an optimal solution to a set cover problem that differs from the initial problem in one position of the constraint matrix does not exist if the optimal solution of the original problem is known and DistNP is not a subset of Average-P. A similar result takes place for the knapsack problem.     

基于遗传算法的物流配送路径优化问题研究

遗传算法是一种基于自然进化原理的全局搜索随机算法。遗传算法在选址问题、配送问题、调度问题、运输问题、布局问题方面意义重大。在建立物流配送路径优化问题数学模型的基础上,构造了求解该问题的遗传算法。该遗传算法采用常用的二进制编码,在个体选择上结合使用最优个体保留策略和轮盘赌法。最后以这种方法进行了实验计算,通过计算结果表明,用遗传算法进行物流配送路径优化,可以方便有效地求得问题的最优解或近似最优解。     

实例序列条数为常数9的翻转距离星树问题

讨论翻转距离星树问题，证明实例中有向符号序列个数为9时，翻转距离星树问题是NP-难解问题，并给出了一个该问题的多项式时间近似算法．     

Constrained Optimal Hybrid Control of a Flow Shop System

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem.     

限制树宽的图的最小标记生成数算法

本文研究了图的最小标记生成树问题。首先介绍在一般图上基于搜索树的最小标记生成树的算法;然后考虑了限制树宽的图,得到了效率更高的算法。该算法在树宽为常数的情况下,时间复杂度关于图的顶点个数为多项式,从而也证明了最小标记生成树在限制树宽的图上属于确定参数可解问题。     

The complexity of single machine scheduling with two distinct deadlines and identical decreasing rates of processing times

The makespan problem on a single machine for a set of tasks with two distinct deadlines and identical decreasing rates of processing times is considered in this paper. Ho et al. [1] have proposed a model of a task system in which the processing time of a task decreases with its starting time. When the decreasing rate is identical, the computational complexity of the makespan problem with two distinct deadlines is posed as an open problem. In this paper we show that the problem is NP-complete. It follows that both the corresponding flow time problem and maximum lateness problem are also NP-complete.     

Characterization of Problem Stores

This paper introduces the concept of problem stores: static stores whose dependent loads often miss in the cache. Accurately identifying problem stores allows the early determination of addresses likely to cause later misses, potentially allowing for the development of novel, proactive prefetching and memory hierarchy management schemes. We present a detailed empirical characterization of problem stores using the SPEC2000 CPU benchmarks. The data suggests several key observations about problem stores. First, we find that the number of important problem stores is typically quite small; the worst 100 problem stores write the values that will lead to about 90% of non-cold misses for a variety of cache configurations. We also find that problem stores only account for 1 in 8 dynamic stores, though they result in 9 of 10 misses. Additionally, the problem stores’ dependent loads miss in the L2 cache a larger fraction of the time than loads not dependent on problem stores. We also observe the set of problem stores is stable across a variety of cache configurations. Finally, we found that the instruction distance from problem store to miss and problem store to evict is often greater than one million instructions, but the value is often needed within 100,000 instructions of the eviction.     

The levenberg-marquardt method for approximation of solutions of irregular operator equations

An ill-posed problem is considered in the form of a nonlinear operator equation with a discontinuous inverse operator. It is known that in investigating a high convergence of the methods of the type of Levenberg-Marquardt (LM) method, one is forced to impose very severe constraints on the problem operator. In the suggested article the LM method convergence is set up not for the initial problem, but for the Tikhonov-regularized equation. This makes it possible to construct a stable Fejer algorithm for approximation of the solution of the initial irregular problem at the conventional, comparatively nonburdensome conditions on the operator. The developed method is tested on the solution of an inverse problem of geophysics.     

On a class of branching problems in broadcasting and distribution

We introduce the following network optimization problem: given a directed graph with a cost function on the arcs, demands at the nodes, and a single source s, find the minimum cost connected subgraph from s such that its total demand is no less than lower bound D. We describe applications of this problem to disaster relief and media broadcasting, and show that it generalizes several well-known models including the knapsack problem, the partially ordered knapsack problem, the minimum branching problem, and certain scheduling problems. We prove that our problem is strongly NP-complete and give an integer programming formulation. We also provide five heuristic approaches, illustrate them with a numerical example, and provide a computational study on both small and large sized, randomly generated problems. The heuristics run efficiently on the tested problems and provide solutions that, on average, are fairly close to optimal.     

蚁群算法一阶欺骗性问题的时间复杂度分析

文中研究蚁群算法求解欺骗性问题时的时间复杂度。以蚁群算法一阶欺骗性问题n-bit陷阱问题为例, 证明使用信息素带限的最大最小蚁群算法求解n-bit陷阱问题达到最优解的时间复杂度为O(n²mlnn),其中n为问题的规模,m为蚂蚁的个数。实验结果验证上述结论的正确性。     

多一次Paillier求逆问题与并发安全的鉴别方案

从计算难解性的角度重新考察Paillier的陷门单向函数,并提出多一次Paillier求逆问题这一关于Paillier求逆问题的推广问题.从计算难解性的角度考察了多一次Paillier求逆问题与Bellare等人提出的多一次RSA求逆问题之间的关系,并证明了在计算难解性的意义上。多一次Paillier求逆问题等价于多一次RSA求逆问题.以此为基础,进而提出一种新的鉴别方案,并证明在多一次Paillier求逆问题的难解性假设下这一鉴别方案具备并发安全性.     

Bounded coloring of co-comparability graphs and the pickup and delivery tour combination problem

The Double Traveling Salesman Problem with Multiple Stacks is a vehicle routing problem in which pickups and deliveries must be performed in two independent networks. The items are stored in stacks and repacking is not allowed. Given a pickup and a delivery tour, the problem of checking if there exists a valid distribution of items into s stacks of size h that is consistent with the given tours, is known as Pickup and Delivery Tour Combination (PDTC) problem.In the paper, we show that the PDTC problem can be solved in polynomial time when the number of stacks s is fixed but the size of each stack is not. We build upon the equivalence between the PDTC problem and the bounded coloring (BC) problem on permutation graphs: for the latter problem, s is the number of colors and h is the number of vertices that can get a same color. We show that the BC problem can be solved in polynomial time when s is a fixed constant on co-comparability graphs, a superclass of permutation graphs. To the contrary, the BC problem is known to be hard on permutation graphs when h≥6 is a fixed constant, but s is unbounded.     

Conversion of the Steiner Problem on the Euclidean Plane to the Steiner Problem on Graph

If in the Steiner problem on graph the number of terminal nodes is much smaller than that of all graph nodes (say 50 against 1000), then one can see in its solution (the minimum tree) a system of paths on the graph connecting the terminal vertices, rather than a collection of separate edges (or arcs). This path system is similar to the system of segments making up the Steiner tree on the Euclidean plane: the local degree of the terminal nodes usually is 1, and the local degree of some nodes making up the paths is 3 or more. These nodes are the counterparts of the Steiner points in the problem on plane. Structural similarity of the trees in the Steiner problems on the Euclidean plane and on graph enables one to construct the algorithm to solve the Steiner problem on graph along the same lines as in the Steiner problem on the Euclidean plane. On the other hand, the solution of a problem on graph may be regarded as that on the Euclidean plane, provided that the graph satisfies certain requirements.     

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。