Simple Approximation Algorithm Research Articles

Rapid epistatic mixed-model association studies by controlling multiple polygenic effects.

We have developed a rapid mixed model algorithm for exhaustive genome-wide epistatic association analysis by controlling multiple polygenic effects. Our model can simultaneously handle additive by additive epistasis, dominance by dominance epistasis and additive by dominance epistasis, and account for intrasubject fluctuations due to individuals with repeated records. Furthermore, we suggest a simple but efficient approximate algorithm, which allows the examination of all pairwise interactions in a remarkably fast manner of linear with population size. Simulation studies are performed to investigate the properties of REMMAX. Application to publicly available yeast and human data has showed that our mixed model-based method has similar performance with simple linear model on computational efficiency. It took less than 40 h for the pairwise analysis of 5000 individuals genotyped with roughly 350000 SNPs with five threads on Intel Xeon E5 2.6 GHz CPU. Source codes are freely available at https://github.com/chaoning/GMAT. Supplementary data are available at Bioinformatics online.

GREEDY SHORTEST SUPERSTRING WITH DELAYED RANDOM CHOICE

The shortest superstring problem for a given set of strings is to find a string of minimum length such that each input string is a substring of the resulting string. This problem is known to be NP-complete. A simple and popular approximation algorithm for this problem is GREEDY, which at each step merges a pair of strings that have maximum overlap. If more than one pair have maximum overlap, it takes a pair in random. In this paper, we modify GREEDY such that instead of taking a pair in random, it takes the pair for which the overlap in the next step is maximum. We analyze our algorithm and compare it with GREEDY for different types of input. We implement both the algorithms and the experimental results show that our algorithm can outperform GREEDY substantially in many cases, and in general our algorithm is same or better than GREEDY.

A Novel Fast Parallel Batch Scheduling Algorithm for Solving the Independent Job Problem

With the rapid economic development, manufacturing enterprises are increasingly using an efficient workshop production scheduling system in an attempt to enhance their competitive position. The classical workshop production scheduling problem is far from the actual production situation, so it is difficult to apply it to production practice. In recent years, the research on machine scheduling has become a hot topic in the fields of manufacturing systems. This paper considers the batch processing machine (BPM) scheduling problem for scheduling independent jobs with arbitrary sizes. A novel fast parallel batch scheduling algorithm is put forward to minimize the makespan in this paper. Each of the machines with different capacities can only handle jobs with sizes less than the capacity of the machine. Multiple jobs can be processed as a batch simultaneously on one machine only if their total size does not exceed the machine capacity. The processing time of a batch is determined by the longest of all the jobs processed in the batch. A novel and fast 4.5-approximation algorithm is developed for the above scheduling problem. For the special case of all the jobs having the same processing times, a simple and fast 2-approximation algorithm is achieved. The experimental results show that fast algorithms further improve the competitive ratio. Compared to the optimal solutions generated by CPLEX, fast algorithms are capable of generating a feasible solution within a very short time. Fast algorithms have less computational costs.

Targeted optimal-path problem for electric vehicles with connected charging stations.

Path planning for electric vehicles (EVs) can alleviate the limited cruising range and “range anxiety”. Many existing path optimization models cannot produce satisfactory solutions due to the imposition of too many assumptions and simplifications. The targeted optimal-path problem for electric vehicles (EV-TOP), which is proposed in the paper, aims at identifying the targeted optimal path for EVs under the limited battery level. It minimizes the travel cost, which is composed of the travel time and the total time that is spent at charging stations (CSs). The model is much more realistic due to the prediction and the consideration of the waiting times at CSs and more accurate approximations of the electricity consumption function and the charging function. Charging station information and the road traffic state are utilized to calculate the travel cost. The EV-TOP is decomposed into two subproblems: a constrained optimal path problem in the network (EV1-COP) and a shortest path problem in the meta-network (EV2-SP). To solve the EV1-COP, the Lagrangian relaxation algorithm, the simple efficient approximation (SEA) algorithm, and the Martins (MS) deletion algorithm are used. The EV2-SP is solved using Dijkstra’s algorithm. Thus, a polynomial-time approximation algorithm for the EV-TOP is developed. Finally, two computational studies are presented. The first study assesses the performance of the travel cost method. The second study evaluates the performance of our EV-TOP by comparing it with a well-known method.

On Approximating the Stationary Distribution of Time-Reversible Markov Chains

Approximating the stationary probability of a state in a Markov chain through Markov chain Monte Carlo techniques is, in general, inefficient. Standard random walk approaches require $\tilde {O}(\tau /\pi (v))$ operations to approximate the probability π(v) of a state v in a chain with mixing time τ, and even the best available techniques still have complexity $\tilde {O}(\tau ^{1.5}/\pi (v)^{0.5})$; and since these complexities depend inversely on π(v), they can grow beyond any bound in the size of the chain or in its mixing time. In this paper we show that, for time-reversible Markov chains, there exists a simple randomized approximation algorithm that breaks this “small-π(v) barrier”.

On the Performance of a Simple Approximation Algorithm for the Longest Path Problem

The longest path problem is known to be NP-hard. Moreover, they cannot be approximated within a constant ratio, unless ${\rm P=NP}$. The best known polynomial time approximation algorithms for this problem essentially find a path of length that is the logarithm of the optimum.In this paper we investigate the performance of an approximation algorithm for this problem in almost every case. We show that a simple algorithm, based on depth-first search, finds on almost every undirected graph $G=(V,E)$ a path of length more than $|V|-3\sqrt{|V| \log |V|}$ and so has performance ratio less than $1+4\sqrt{\log |V|/|V|}$.\

A simple approximation algorithm for the diameter of a set of points in an Euclidean plane.

Approximation algorithms with linear complexities are required in the treatments of big data, however, present algorithms cannot output the diameter of a set of points with arbitrary accuracy and near-linear complexity. By introducing the partition technique, we introduce a very simple approximation algorithm with arbitrary accuracy ε and a complexity of O(N + ε−1 log ε−1) for the cases that all points are located in an Euclidean plane. The error bounds are proved strictly, and are verified by numerical tests. This complexity is better than existing algorithms, and the present algorithm is also very simple to be implemented in applications.

Near-optimal control of nonlinear systems with simultaneous controlled and random switches

Abstract We consider dual switched systems, in which two switching signals act simultaneously to select the dynamical mode. The first signal is controlled and the second is random, with probabilities that evolve either periodically or as a function of the dwell time. We formalize both cases as Markov decision processes, which allows them to be solved with a simple approximate dynamic programming algorithm. We illustrate the framework in a problem where the random signal is a delay on the control channel that is used to send the controlled signal to the system.

A linear time randomized approximation algorithm for Euclidean matching

We study the problem of computing Euclidean minimum weight matching which investigates the minimum weight perfect matching in a complete geometric graph on a set of 2n points in the plane. We propose a simple randomized approximation algorithm based on the geometrical aspect of the problem with O(n) expected time. The proposed algorithm computes a matching within at most 3 factors of the optimal solution. We also do some experimental tests to evaluate the performance of the proposed algorithm which indicate the efficiency of the proposed algorithm in finding the approximate matching in the practice.

Improving the Betweenness Centrality of a Node by Adding Links

Betweenness is a well-known centrality measure that ranks the nodes according to their participation in the shortest paths of a network. In several scenarios, having a high betweenness can have a positive impact on the node itself. Hence, in this article, we consider the problem of determining how much a vertex can increase its centrality by creating a limited amount of new edges incident to it. In particular, we study the problem of maximizing the betweenness score of a given node—Maximum Betweenness Improvement (MBI)—and that of maximizing the ranking of a given node—Maximum Ranking Improvement (MRI). We show that MBI cannot be approximated in polynomial-time within a factor (1−1/2e) and that MRI does not admit any polynomial-time constant factor approximation algorithm, both unless P = NP . We then propose a simple greedy approximation algorithm for MBI with an almost tight approximation ratio and we test its performance on several real-world networks. We experimentally show that our algorithm highly increases both the betweenness score and the ranking of a given node and that it outperforms several competitive baselines. To speed up the computation of our greedy algorithm, we also propose a new dynamic algorithm for updating the betweenness of one node after an edge insertion, which might be of independent interest. Using the dynamic algorithm, we are now able to compute an approximation of MBI on networks with up to 10 5 edges in most cases in a matter of seconds or a few minutes.

Delta DLP 3-D Printing of Large Models

This paper presents a 3-D printing system that uses a low-cost off-the-shelf consumer projector to fabricate large models. Compared with traditional digital light processing (DLP) 3-D printers using a single vertical carriage, the platform of our DLP 3-D printer using delta mechanism can also move horizontally in the plane. We show that this system can print 3-D models much larger than traditional DLP 3-D printers. The major challenge to realize 3-D printing of large models in our system comes from how to cover a planar polygonal domain by a minimum number of rectangles with fixed size, which is NP-hard. We propose a simple yet efficient approximation algorithm to solve this problem. The key idea is to segment a polygonal domain using its medial axis and afterward merge small parts in the segmentation. Given an arbitrary polygon $\Omega $ with $n$ generators (i.e., line segments and reflex vertices in $\Omega $ ), we show that the time complexity of our algorithm is $O(n^{2}\log ^{2}~n)$ and the number of output rectangles covering $\Omega $ is $O(Kn)$ , where $K$ is an input-polygon-dependent constant. A physical prototype system is built and several large 3-D models with complex geometric structures have been printed as examples to demonstrate the effectiveness of our approach. Note to Practitioners —Low-cost 3-D printers and 3-D printing of large models are two important but often conflicting goals in manufacturing industry. Usually low-cost devices such as DLP 3-D printer using an off-the-shelf consumer projector cannot print large models, due to the small projection area of the projector. In this paper, we propose to horizontally move the platform such that a large area can be printed by the composition of multiple small projection areas. Based on this new mechanism, we propose a simple yet efficient algorithm to cover an arbitrary polygonal shape (possibly with holes or multiple disjoint polygons) by a small set of rectangles with fixed size. Our algorithm is theoretically sound and can be easily implemented. We built a physical prototype system of the proposed low-cost Delta DLP 3-D printer, which successfully prints several large models with satisfied mechanical properties.

On scenario aggregation to approximate robust combinatorial optimization problems

As most robust combinatorial min–max and min–max regret problems with discrete uncertainty sets are NP-hard, research in approximation algorithm and approximability bounds has been a fruitful area of recent work. A simple and well-known approximation algorithm is the midpoint method, where one takes the average over all scenarios, and solves a problem of nominal type. Despite its simplicity, this method still gives the best-known bound on a wide range of problems, such as robust shortest path or robust assignment problems. In this paper, we present a simple extension of the midpoint method based on scenario aggregation, which improves the current best K-approximation result to an $$(\varepsilon K)$$ -approximation for any desired $$\varepsilon > 0$$ . Our method can be applied to min–max as well as min–max regret problems.

The multi-service center problem

We propose a new type of network location problem for placing multiple but distinct facilities, called the p-service center problem. In this problem, we are to locate p facilities in the graph, each of which provides distinct service required by all vertices. For each vertex, its p-service distance is the summation of its weighted distances to the p facilities. The objective is to minimize the maximum value among the p-service distances of all vertices.In this paper, we show that the p-service center problem on a general graph is NP-hard, and propose a simple approximation algorithm of factor p/c for any integer constant c. Moreover, we study the basic case p=2 on paths and trees. When the underlying network is a path, we solve the 2-service center problem in O(n) time, where n is the number of vertices. When the underlying network is a tree, we give an O(n3)-time algorithm for the case of nonnegative weights, an O(nlog⁡n)-time algorithm for the case of positive weights, and an O(n)-time algorithm for the case of uniform weights.

Fast Approximation Algorithms for the One-Warehouse Multi-Retailer Problem Under General Cost Structures and Capacity Constraints

We consider a well-studied multi-echelon (deterministic) inventory control problem, known in the literature as the one-warehouse multi-retailer (OWMR) problem. We propose a simple and fast 2-approximation algorithm for this NP-hard problem, by recombining the solutions of single-echelon relaxations at the warehouse and at the retailers. We then show that our approach remains valid under quite general assumptions on the cost structures and under capacity constraints at some retailers. In particular, we present the first approximation algorithms for the OWMR problem with nonlinear holding costs, truckload discount on procurement costs, or with capacity constraints at some retailers. In all cases, the procedure is purely combinatorial and can be implemented to run in low polynomial time.

Simple Approximation Algorithms for Balanced MAX 2SAT

We study simple algorithms for the balanced MAX 2SAT problem, where we are given weighted clauses of length one and two with the property that for each variable x the total weight of clauses that x appears in equals the total weight of clauses for $$\overline{x}$$ . We show that such instances have a simple structural property in that any optimal solution can satisfy at most the total weight of the clauses minus half the total weight of the unit clauses. Using this property and a novel analysis of the computation tree, we are able to show that a large class of greedy algorithms, including Johnson’s algorithm, gives a $$\frac{3}{4}$$ -approximation algorithm for balanced MAX 2SAT; a similar statement is false for general MAX 2SAT instances. We further give a spectral 0.81-approximation algorithm for balanced MAX E2SAT instances (in which each clause has exactly 2 literals) by a reduction to a spectral algorithm of Trevisan for the maximum colored cut problem. We provide experimental results showing that this spectral algorithm performs well and is slightly better than Johnson’s algorithm and the Goemans–Williamson semidefinite programming algorithm on balanced MAX E2SAT instances.

A simple approximation algorithm for minimum weight partial connected set cover

This paper studies the minimum weight partial connected set cover problem (PCSC). Given an element set U, a collection \({\mathcal {S}}\) of subsets of U, a weight function c : \({\mathcal {S}} \rightarrow {\mathbb {Q}}^{+}\), a connected graph \(G_{{\mathcal {S}}}\) on vertex set \({\mathcal {S}}\), and a positive integer \(k\le |U|\), the goal is to find a minimum weight subcollection \({\mathcal {S}}' \subseteq {\mathcal {S}}\) such that the subgraph of G induced by \({\mathcal {S}}'\) is connected, \(|\bigcup _{S\in {\mathcal {S}}^{\prime }}S| \ge k\), and the weight of \({\mathcal {S}}'\) is minimum. If the graph \(G_{{\mathcal {S}}}\) has the property that any two sets with a common element has hop distance at most r in \(G_{{\mathcal {S}}}\), then the problem is called r-hop PCSC. We presented an \(O(\ln (m+n))\)-approximation algorithm for the minimum weight 1-hop PCSC problem and an \(O(\ln (m+n))\)-approximation algorithm for the minimum cardinality r-hop PCSC problem. Our performance ratio improves previous results and our method is much simpler than previous methods.

Simpler and Better Approximation Algorithms for the Unweighted Minimum Label s-t Cut Problem

Given a graph $$G=(V,E)$$ with a label set $$L = \{\ell _1, \ell _2, \ldots , \ell _q \}$$ , in which each edge has a label from L, and a source $$s \in V$$ together with a sink $$t \in V$$ , the Minimum Label s-t Cut problem asks to pick a set $$L' \subseteq L$$ of labels with minimized cardinality, such that the removal of all edges with labels in $$L'$$ from G disconnects s and t. Let $$n = |V|$$ and $$m = |E|$$ . The previous best known approximation ratio for this problem in literature is $$O(m^{1/2})$$ . We present two simple and purely combinatorial approximation algorithms for the problem with ratios $$O(n^{2/3}/\text {OPT}^{1/3})$$ and $$O(m^{1/2} / \text {OPT}^{1/2})$$ , where $$\text {OPT}$$ is the optimal value of the input instance. The former result gives the first approximation ratio which is sublinear in n for the problem, and in particular applies to the instances with dense graphs (e.g., $$m = \varTheta (n^2)$$ ). The latter result improves the previous ratio $$O(m^{1/2})$$ , as we can always assume that $$\text {OPT}$$ is a super-constant.

Energetic electron processes fluorescence effects for structured nanoparticles X-ray analysis and nuclear medicine applications

Superparamagnetic iron oxide (SPIO) nanoparticles are widely used as contrast agents for nuclear magnetic resonance imaging (MRI), and can be modified for improved imaging or to become tissue-specific or even protein-specific. The knowledge of their detailed elemental composition characterisation and potential use in nuclear medicine applications, is, therefore, an important issue. X-ray fluorescence techniques such as particle induced X-ray emission (PIXE) or X-ray fluorescence spectrometry (XRF), can be used for elemental characterisation even in problematic situations where very little sample volume is available. Still, the fluorescence coefficient of Fe is such that, during the decay of the inner-shell ionised atomic structure, keV Auger electrons are produced in excess to X-rays. Since cross-sections for ionisation induced by keV electrons, for low atomic number atoms, are of the order of 103 barn, care should be taken to account for possible fluorescence effects caused by Auger electrons, which may lead to the wrong quantification of elements having atomic number lower than the atomic number of Fe. Furthermore, the same electron processes will occur in iron oxide nanoparticles containing 57Co, which may be used for nuclear medicine therapy purposes. In the present work, simple approximation algorithms are proposed for the quantitative description of radiative and non-radiative processes associated with Auger electrons cascades. The effects on analytical processes and nuclear medicine applications are quantified for the case of iron oxide nanoparticles, by calculating both electron fluorescence emissions and energy deposition on cell tissues where the nanoparticles may be embedded.

A Constant Factor Approximation Algorithm for Fault-Tolerant k -Median

In this article, we consider the fault-tolerant k -median problem and give the first constant factor approximation algorithm for it. In the fault-tolerant generalization of the classical k -median problem, each client j needs to be assigned to at least r j ⩾ 1 distinct open facilities. The service cost of j is the sum of its distances to the r j facilities, and the k -median constraint restricts the number of open facilities to at most k . Previously, a constant factor was known only for the special case when all r j s are the same, and alogarithmic approximation ratio was known for the general case. In addition, we present the first polynomial time algorithm for the fault-tolerant k -median problem on a path or an HST by showing that the corresponding LP always has an integral optimal solution. We also consider the fault-tolerant facility location problem, in which the service cost of j can be a weighted sum of its distance to the r j facilities. We give a simple constant factor approximation algorithm, generalizing several previous results that work only for nonincreasing weight vectors.

On the Maximum Betweenness Improvement Problem

The betweenness is a well-known measure of centrality of a node in a network. We consider the problem of determining how much a node can increase its betweenness centrality by creating a limited amount of new edges incident to it. If the graph is directed, this problem does not admit a polynomial-time approximation scheme (unless P=NP) and a simple greedy approximation algorithm guarantees an almost tight approximation ratio [E. D. Demaine and M. Zadimoghaddam. Minimizing the diameter of a network using shortcut edges. In Proc. of the 12th Scandinavian Symp. and Work. on Algorithm Theory (SWAT), volume 6139 of LNCS, pages 420–431. Springer, 2010].In this paper we focus on the undirected graph case: we show that also in this case the problem does not admit a polynomial-time approximation scheme (unless P=NP). Moreover, we show that, differently from the directed case, the greedy algorithm can have an unbounded approximation ratio. In order to test the practical performance of the greedy algorithm, we experimentally measured its efficiency in term of ranking improvement, comparing it with another algorithm that simply adds edges to the nodes that have highest betweenness. Our experiments show that the greedy algorithm adds only few edges in order to increase the betweenness of a node and to reach the top positions in the ranking. Moreover, the greedy algorithm outperforms the second approach.