Minimum cost flows with minimum quantities

We consider a variant of the well-known minimum cost flow problem where the flow on each arc in the network is restricted to be either zero or above a given lower bound. The problem was recently shown to be weakly NP-complete even on series-parallel graphs. We start by showing that the problem is strongly NP-complete and cannot be approximated in polynomial time (unless P=NP) up to any polynomially computable function even when the graph is bipartite and the given instance is guaranteed to admit a feasible solution. Moreover, we present a pseudo-polynomial-time exact algorithm and a fully polynomial-time approximation scheme (FPTAS) for the problem on series-parallel graphs.     

Bounded parallel-batch scheduling on single and multi machines for deteriorating jobs

We consider the bounded parallel-batch scheduling problem in which the processing time of a job is a simple linear function of its starting time. The objective is to minimize the makespan. When the jobs have identical release dates, we present an optimal algorithm for the single-machine problem and an fully polynomial-time approximation scheme for the parallel-machine problem. When the jobs have distinct release dates, we show that the single-machine problem is NP-hard and present an optimal algorithm for one special case.     

Generating the fewest redundancy-free scheme trees from acyclic conceptual-model hypergraphs in polynomial time

Generating the fewest redundancy-free scheme trees from conceptual-model hypergraphs is NP-hard [17]. We show, however, that the problem has a polynomial-time solution if the conceptual-model hypergraph is acyclic. We define conceptual-model hypergraphs, cycles, and scheme trees, and then present a polynomial-time algorithm and show that it generates the fewest redundancy-free scheme trees. As a practical application for the algorithm, we comment on its use for the design of “good” XML schemas for data storage.     

Randomized approximation scheme for resource allocation in hybrid-cloud environment

Using the virtually unlimited resource capacity of public cloud, dynamic scaling out of large-scale applications is facilitated. A critical question arises practically here is how to run such applications effectively in terms of both cost and performance. In this paper, we explore how resources in the hybrid-cloud environment should be used to run Bag-of-Tasks applications. Having introduced a simple yet effective objective function, our algorithm helps the user to make a better decision for realization of his/her goal. Then, we cope with the problem in two different cases of “known” and “unknown” running time of available tasks. A solution to approximate the optimal value of user’s objective function will be provided for each case. Specifically, a fully polynomial-time randomized approximation scheme based on a Monte Carlo sampling method will be presented in case of unknown running time. The experimental results confirm that our algorithm approximates the optimal solution with a little scheduling overhead.     

Defining static floor field of evacuation model in large exit scenario

The floor field model has been widely used to study pedestrian movement. But the traditional methods of setting the static floor field would lead to highly insufficient utilization of the exit region when the exit width is very large. In order to solve the problem and to study the utilization of wide exit, in this paper we put forward an idea of “virtual reference point” and propose a new method of building static floor field. A virtual reference point can be regarded as a point sink of the floor field, it has minimum field value. The position of the virtual reference point decides the distribution of static floor field values in the model. We further explore the relationship between the virtual reference point position and the exit width using regression analysis. It seems that a proper position of virtual reference point will make the exit be fully used and get high evacuation efficiency. We then analyze how to lead people to fully utilize the exit by changing the configuration of exit guidance, and give a primary scheme.     

Bounded single-machine parallel-batch scheduling with release dates and rejection

We consider the bounded single-machine parallel-batch scheduling problem with release dates and rejection. A job is either rejected, in which case a certain penalty has to be paid, or accepted and then processed on the machine. The objective is to minimize the sum of the makespan of the accepted jobs and the total penalty of the rejected jobs. When the jobs have identical release dates, we present a polynomial-time algorithm. When the jobs have a constant number of release dates, we give a pseudo-polynomial-time algorithm. For the general problem, we provide a 2-approximation algorithm and a polynomial-time approximation scheme.     

Two-phase evacuation route planning approach using combined path networks for buildings and roads

This paper addresses the problem of modeling evacuation routes from a building and out of an affected area. The evacuation route involves pathways such as corridors, and stairs in buildings and road networks and sidewalks outside the building. To illustrate such an approach, we consider the problem of finding evacuation paths from an urban building and out of a predetermined neighborhood of the building on foot. A case study for a college campus building and small set of road around it is provided. There are a pre-defined set of exit points out of the target building and out of the road network serving the building. A two-step approach with an uncapacitated network model for route finding and a capacitated scheduling algorithm for evacuation time computation is proposed. A recent efficient heuristic algorithm is selected as a reference for comparative analysis. The process of creating a combined building and road path network data is discussed. The key results are the competitive evacuation time provided by the proposed uncapacitated route planning model, simple pedestrian flow capacity formulas for corridors and roads from readily available geometric data, and the illustration of the creation and use of combined building and road path network.     

Opaque Sets

The problem of finding “small” sets that meet every straight-line which intersects a given convex region was initiated by Mazurkiewicz in 1916. We call such a set an opaque set or a barrier for that region. We consider the problem of computing the shortest barrier for a given convex polygon with n vertices. No exact algorithm is currently known even for the simplest instances such as a square or an equilateral triangle. For general barriers, we present an approximation algorithm with ratio $\frac{1}{2}+ \frac{2 +\sqrt{2}}{\pi}=1.5867\ldots$ ?. For connected barriers we achieve the approximation ratio 1.5716, while for single-arc barriers we achieve the approximation ratio $\frac{\pi+5}{\pi+2} = 1.5834\ldots$ ?. All three algorithms run in O(n) time. We also show that if the barrier is restricted to the (interior and the boundary of the) input polygon, then the problem admits a fully polynomial-time approximation scheme for the connected case and a quadratic-time exact algorithm for the single-arc case.     

Call Control in Rings

The call control problem is an important optimization problem encountered in the design and operation of communication networks. The goal of the call control problem in rings is to compute, for a given ring network with edge capacities and a set of paths in the ring, a maximum cardinality subset of the paths such that no edge capacity is violated. We give a polynomial-time algorithm to solve the problem optimally. The algorithm is based on a decision procedure that checks whether a solution with at least k paths exists, which is in turn implemented by an iterative greedy approach operating in rounds. We show that the algorithm can be implemented efficiently and, as a by-product, obtain a linear-time algorithm to solve the problem in chains optimally. For the weighted version of call control in rings, where each path is associated with a weight and the goal is to maximize the total weight of the paths in the solution, we present a simple 2-approximation algorithm and a polynomial-time approximation scheme. While the complexity of the weighted version remains open, we show that it is at least as hard as the bipartite exact matching problem, which has not been resolved to be in P or NP-hard. This latter result follows from recent work by Hochbaum and Levin.     

Lot-sizing scheduling with batch setup times

This paper is concerned with scheduling independent jobs on m parallel machines in such a way that the makespan is minimized. Each job j is allowed to split arbitrarily into several parts, which can be individually processed on any machine at any time. However, a setup for uninterrupted s_j time units is required before any split part of job j can be processed on any machine. The problem is strongly NP-hard if the number m of machines is variable and weakly NP-hard otherwise. We give a polynomial-time -approximation algorithm for the former case and a fully polynomial-time approximation scheme for the latter. AMS Subject Classifications: 68M20 · 90B35 · 90C59     

A random fuzzy minimum spanning tree problem through a possibility-based value at risk model

This paper considers a minimum spanning tree problem under the situation where costs for constructing edges in a network include both fuzziness and randomness. In particular, this article focuses on the case that the edge costs are expressed by random fuzzy variables. A new decision making model based on a possibility measure and a value at risk measure is proposed in order to find a solution which fully reflects random and fuzzy information. It is shown that an optimal solution of the proposed model is obtained by a polynomial-time algorithm.     

Erratum to “Minimum cost flows with minimum quantities” [Information Processing Letters 111 (11) (2011) 533–537]

This note identifies an error in the fully polynomial-time approximation scheme on series-parallel graphs presented in our paper “Minimum cost flows with minimum quantities” which appeared as [Sven O. Krumke, Clemens Thielen, Minimum cost flows with minimum quantities, Information Processing Letters 111 (11) (2011) 533–537]. In fact, we prove inapproximability of the problem on series-parallel graphs and of a related problem, thereby identifying a similar error in a related paper.     

Bounded-Diameter Minimum-Cost Graph Problems

We present approximation algorithms for two closely related bicriteria problems. Given a graph with two weight functions on the edges, length and cost, we consider the Bounded-Diameter Minimum-Cost Steiner Tree (BDMST) problem and the Bounded-Diameter Minimum-Cardinality Edge Addition (BDMC) problem. We present a parameterized approximation algorithm for the BDMST problem with a bicriteria approximation ratio of (O(p log s/log p),O(log s/log p)) where the first factor gives the relaxation on the diameter bound, the second factor is the cost-approximation factor, s is the number of required nodes and p, 1 ≤ p < s, is an input parameter. The parameter p allows a trade-off between the two approximation factors. This is the first improvement of the cost-approximation factor since the scheme proposed by Marathe et al. [9]. For example, p can be set to s^α to obtain an (O(s^α),O(1)) approximation for a constant α. The algorithm is very simple and is suitable for distributed implementations. We also present the first algorithm for Bounded-Hops Minimum-Cost Steiner Tree for complete graphs with triangle inequality. This model includes graphs defined by points in a Euclidean space of any dimension. The algorithm guarantees an approximation ratio of (O(log_ds),O(log_ds)) where d is the bound on the diameter. This is an improvement over the general-case approximation when d is comparable with s. For example, the ratio is (O(1),O(1)) for any d = s^α where α is a constant between 0 and 1. For the case where the number of terminals is a constant and all edge lengths are unit, we have a polynomial-time algorithm. This can be extended to any length function providing a (1 + ε) in the approximation with ε showing up in the time complexity of the algorithm. For another special case, where the cost of any edge is either 1 or 0 and the length of each edge is positive, an algorithm with approximation ratio of (O(log(c log s)), O(log(c log s))) is presented, where c is the cost of the optimal solution. This approximation is a significant improvement over (O(log s),O(log s)) when the cost of the solution c is much smaller than the number of terminals s. This is useful when an existing multicast network is to be modified to accommodate new terminals with the addition of as few new edges as possible. We also propose an approximation algorithm for the Bounded-Diameter Minimum-Cardinality Edge Addition problem, which achieves an O(log n) approximation while relaxing the diameter bound by 2. While this ratio is the same as the one claimed in [3], this algorithm is simple and combinatorial rather than based on the Linear Program solution and can be readily modified for a distributed implementation.     

An approximate A algorithm and its application to the SCS problem

In this paper we deal with algorithm A^* and its application to the problem of finding the shortest common supersequence of a set of sequences. A^* is a powerful search algorithm which may be used to carry out concurrently the construction of a network and the solution of a shortest path problem on it. We prove a general approximation property of A^* which, by building a smaller network, allows us to find a solution with a given approximation ratio. This is particularly useful when dealing with large instances of some problem. We apply this approach to the solution of the shortest common supersequence problem and show its effectiveness.     

虚拟结点法在多出口疏散方案中的适用性分析

城市化进程的迅速推进促使城市规模日益扩大、人口密度不断提高,制定一个有效的应急疏散方案,使得人员在存在危险时快速且有序地撤离到安全场所,已经成为迫切需要解决的问题。目前已有一些研究成果,而虚拟结点法在网络流中已有较多应用。结合基于道路网络的单出口分阶段疏散算法,将虚拟结点法应用在多出口疏散方案中,旨在对虚拟结点法的适用性进行分析。模拟结果显示,该方法能够有序、快速地组织大规模人群到安全场所。     

Approximation algorithms for tree alignment with a given phylogeny

We study the following fundamental problem in computational molecular biology: Given a set of DNA sequences representing some species and a phylogenetic tree depicting the ancestral relationship among these species, compute an optimal alignment of the sequences by the means of constructing a minimum-cost evolutionary tree. The problem is an important variant of multiple sequence alignment, and is widely known astree alignment. We design an efficient approximation algorithm with performance ratio 2 for tree alignment. The algorithm is then extended to a polynomial-time approximation scheme. The construction actually works for Steiner trees in any metric space, and thus implies a polynomial-time approximation scheme for planar Steiner trees under a given topology (with any constant degree). To our knowledge, this is the first polynomial-time approximation scheme in the fields of computational biology and Steiner trees. The approximation algorithms may be useful in evolutionary genetics practice as they can provide a good initial alignment for the iterative method in [23].Supported in part by NSERC Operating Grant OGP0046613.Supported in part by NSERC Operating Grant OGP0046613 and a Canadian Genome Analysis and Technology Research Grant.Supported in part by US Department of Energy Grant DE-FG03-90ER6099.     

There is no EPTAS for two-dimensional knapsack

In the d-dimensional (vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P=NP, there is no fully polynomial-time approximation scheme for d-dimensional knapsack, already for d=2. The best known result is a polynomial-time approximation scheme (PTAS) due to Frieze and Clarke [A.M. Frieze, M. Clarke, Approximation algorithms for the m-dimensional 0-1 knapsack problem: worst-case and probabilistic analyses, European J. Operat. Res. 15 (1) (1984) 100-109] for the case where d?2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS (EPTAS).In this paper we resolve this question by showing that there is no EPTAS for d-dimensional knapsack, already for d=2, unless W[1]=FPT. Furthermore, we show that unless all problems in SNP are solvable in sub-exponential time, there is no approximation scheme for two-dimensional knapsack whose running time is , for any function f. Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist.     

A Polynomial Time Approximation Scheme for Minimum Cost Delay-Constrained Multicast Tree under a Steiner Topology

This paper is concerned with a restricted version of minimum cost delay-constrained multicast in a network where each link has a delay and a cost. Given a source vertex $s$ and $p$ destination vertices $t_1, t_2, \ldots, t_p$ together with $p$ corresponding nonnegative delay constraints $d_1, d_2, \ldots, d_p$, many QoS multicast problems seek a minimum cost multicast tree in which the delay along the unique $s$--$t_i$ path is no more than $d_i$ for $1 \le i \le p$. This problem is NP-hard even when the topology of the multicast tree is fixed. In this paper we show that every multicast tree has an underlying Steiner topology and that every minimum cost delay-constrained multicast tree corresponds to a minimum cost delay-constrained realization of a corresponding Steiner topology. We present a fully polynomial time approximation scheme for computing a minimum cost delay-constrained multicast tree under a Steiner topology. We also present computational results of a preliminary implementation to illustrate the effectiveness of our algorithm and discuss its applications.     

A Polynomial Time Approximation Scheme for Minimum Cost Delay-Constrained Multicast Tree under a Steiner Topology

This paper is concerned with a restricted version of minimum cost delay-constrained multicast in a network where each link has a delay and a cost. Given a source vertex $s$ and $p$ destination vertices $t_1, t_2, \ldots, t_p$ together with $p$ corresponding nonnegative delay constraints $d_1, d_2, \ldots, d_p$, many QoS multicast problems seek a minimum cost multicast tree in which the delay along the unique $s$--$t_i$ path is no more than $d_i$ for $1 \le i \le p$. This problem is NP-hard even when the topology of the multicast tree is fixed. In this paper we show that every multicast tree has an underlying Steiner topology and that every minimum cost delay-constrained multicast tree corresponds to a minimum cost delay-constrained realization of a corresponding Steiner topology. We present a fully polynomial time approximation scheme for computing a minimum cost delay-constrained multicast tree under a Steiner topology. We also present computational results of a preliminary implementation to illustrate the effectiveness of our algorithm and discuss its applications.     

A new learning algorithm with logarithmic performance index for complex-valued neural networks

In a fully complex-valued feed-forward network, the convergence of the Complex-valued Back Propagation (CBP) learning algorithm depends on the choice of the activation function, learning sample distribution, minimization criterion, initial weights and the learning rate. The minimization criteria used in the existing versions of CBP learning algorithm in the literature do not approximate the phase of complex-valued output well in function approximation problems. The phase of a complex-valued output is critical in telecommunication and reconstruction and source localization problems in medical imaging applications. In this paper, the issues related to the convergence of complex-valued neural networks are clearly enumerated using a systematic sensitivity study on existing complex-valued neural networks. In addition, we also compare the performance of different types of split complex-valued neural networks. From the observations in the sensitivity analysis, we propose a new CBP learning algorithm with logarithmic performance index for a complex-valued neural network with exponential activation function. The proposed CBP learning algorithm directly minimizes both the magnitude and phase errors and also provides better convergence characteristics. Performance of the proposed scheme is evaluated using two synthetic complex-valued function approximation problems, the complex XOR problem, and a non-minimum phase equalization problem. Also, a comparative analysis on the convergence of the existing fully complex and split complex networks is presented.