A successive approximation algorithm for the multiple knapsack problem

It is well-known that the multiple knapsack problem is NP-hard, and does not admit an FPTAS even for the case of two identical knapsacks. Whereas the 0-1 knapsack problem with only one knapsack has been intensively studied, and some effective exact or approximation algorithms exist. A natural approach for the multiple knapsack problem is to pack the knapsacks successively by using an effective algorithm for the 0-1 knapsack problem. This paper considers such an approximation algorithm that packs the knapsacks in the nondecreasing order of their capacities. We analyze this algorithm for 2 and 3 knapsack problems by the worst-case analysis method and give all their error bounds.     

A simple greedy approximation algorithm for the minimum connected $$$$-Center problem

In this paper, we consider the connected \(k\)-Center (\(CkC\)) problem, which can be seen as the classic \(k\)-Center problem with the constraint of internal connectedness, i.e., two nodes in a cluster are required to be connected by an internal path in the same cluster. \(CkC\) was first introduced by Ge et al. (ACM Trans Knowl Discov Data 2:7, 2008), in which they showed the \(NP\)-completeness for this problem and claimed a polynomial time approximation algorithm for it. However, the running time of their algorithm might not be polynomial, as one key step of their algorithm involves the computation of an \(NP\)-hard problem. We first present a simple polynomial time greedy-based \(2\)-approximation algorithm for the relaxation of \(CkC\)—the \(CkC^*\). Further, we give a \(6\)-approximation algorithm for \(CkC\).     

An approximation algorithm for the uniform capacitated k-means problem

Journal of Combinatorial Optimization - In this paper, we consider the uniform capacitated k-means problem (UC-k-means), an extension of the classical k-means problem (k-means) in machine learning....     

An approximation algorithm for the maximum spectral subgraph problem

Journal of Combinatorial Optimization - Modifying the topology of a network to mitigate the spread of an epidemic with epidemiological constant $$\lambda $$ amounts to the NP-hard problem of...     

An approximation algorithm for k-center problem on a convex polygon

This paper studies the constrained version of the k-center location problem. Given a convex polygonal region, every point in the region originates a service demand. Our objective is to place k facilities lying on the region’s boundary, such that every point in that region receives service from its closest facility and the maximum service distance is minimized. This problem is equivalent to covering the polygon by k circles with centers on its boundary which have the smallest possible radius. We present an 1.8841-approximation polynomial time algorithm for this problem.     

Heuristics for the network design problem with connectivity requirements

We consider the NP-complete problem of finding a spanning \(k\)-tree of minimum weight in a complete weighted graph. This problem has a number of applications in designing reliable backbone telecommunication networks. We propose effective algorithms based on a greedy strategy and several variable neighborhood search metaheuristics. We also develop an integer linear programming model for calculating a lower bound. Preliminary numerical experiments using random and real-word data sets are reported to show the effectiveness of our approach. In addition, we compare our approach with known metaheuristics.     

An improved approximation algorithm for uncapacitated facility location problem with penalties

In this paper, we consider an interesting variant of the classical facility location problem called uncapacitated facility location problem with penalties (UFLWP for short) in which each client is either assigned to an opened facility or rejected by paying a penalty. The UFLWP problem has been effectively used to model the facility location problem with outliers. Three constant approximation algorithms have been obtained (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001; Jain et al. in J. ACM 50(6):795–824, 2003; Xu and Xu in Inf. Process. Lett. 94(3):119–123, 2005), and the best known performance ratio is 2. The only known hardness result is a 1.463-inapproximability result inherited from the uncapacitated facility location problem (Guha and Khuller in J. Algorithms 31(1):228–248, 1999). In this paper, We present a 1.8526-approximation algorithm for the UFLWP problem. Our algorithm significantly reduces the gap between known performance ratio and the inapproximability result. Our algorithm first enhances the primal-dual method for the UFLWP problem (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001) so that outliers can be recognized more efficiently, and then applies a local search heuristic (Charikar and Guha in Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pp. 378–388, 1999) to further reduce the cost for serving those non-rejected clients. Our algorithm is simple and can be easily implemented. The research of this work was supported in part by NSF through CAREER award CCF-0546509 and grant IIS-0713489. A preliminary version of this paper appeared in the Proceedings of the 11th Annual International Computing and Combinatorics Conference (COCOON’05).     

A new three-machine shop scheduling: complexity and approximation algorithm

The paper investigates a new three-machine shop scheduling problem that arises from many production systems, such as the garment assembly line, etc. In such scenarios, each job consists of three operations, each of which has to be non-preemptively processed by one specific machine. In contrast with the classical three-machine shop scheduling, the processing order of the operations of each job is partially restricted. In particular, the first two operations are ordered and all the same for all jobs, while the third operation is not restricted. The objective is to minimize the makespan. We show the problem is NP-hard in the ordinary sense and present a polynomial time approximation algorithm with a worst case performance ratio of $\frac{3}{2}$ .     

Exact approaches for the directed network design problem with relays

We study the directed network design problem with relays (DNDPR) whose aim is to construct a minimum cost network that enables the communication of a given set of origin-destination pairs. Thereby, expensive signal regeneration devices need to be placed to cover communication distances exceeding a predefined threshold. Applications of the DNDPR arise in telecommunications and transportation. We propose two new integer programming formulations for the DNDPR. The first one is a flow-based formulation with a pseudo-polynomial number of variables and constraints and the second is a cut-based formulation with an exponential number of constraints. Fractional distance values are handled efficiently by augmenting both models with an exponentially-sized set of infeasible path constraints. We develop branch-and-cut algorithms and also consider valid inequalities to strengthen the obtained dual bounds and to speed up convergence. The results of our extensive computational study on diverse sets of benchmark instances show that our algorithms outperform the previous state-of-the-art method based on column generation.     

An approximation algorithm for stochastic multi-level facility location problem with soft capacities

Journal of Combinatorial Optimization - Facility location problem is one of the most important problems in the combinatorial optimization. The multi-level facility location problem and the facility...     

A primal-dual approximation algorithm for the Asymmetric Prize-Collecting TSP

We present a primal-dual ?log(n)?-approximation algorithm for the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. The previous algorithm for the problem (V.H. Nguyen and T.T Nguyen in Int. J. Math. Oper. Res. 4(3):294–301, 2012) which is not combinatorial, is based on the Held-Karp relaxation and heuristic methods such as the Frieze et al.’s heuristic (Frieze et al. in Networks 12:23–39, 1982) or the recent Asadpour et al.’s heuristic for the ATSP (Asadpour et al. in 21st ACM-SIAM symposium on discrete algorithms, 2010). Depending on which of the two heuristics is used, it gives respectively 1+?log(n)? and $3+ 8\frac{\log(n)}{\log(\log(n))}$ as an approximation ratio. Our algorithm achieves an approximation ratio of ?log(n)? which is weaker than $3+ 8\frac{\log(n)}{\log(\log(n))}$ but represents the first combinatorial approximation algorithm for the Asymmetric Prize-Collecting TSP.     

弹性需求下网络设计问题和电子路票问题研究

应用双(二)层规划模型研究弹性需求下网络设计问题与电子路票收取问题,其中只考虑在部分路段进行路段能力扩充和收取电子路票.上层决策者(网络规划者)选择路段能力增加和收取电子路票的数量来获得最优的社会总福利.下层决策者(网络用户)选择路径来最小化他们的出行成本(路径出行时间与所付出电子路票的价值的和).应用下层规划问题的Ka-rush-Kuhn-Tucker(KKT)条件,将双层规划模型转化为单层规划模型.为了解决互补条件所造成的求解困难,本文构造了松弛算法进行求解,并用数值试验研究了模型和算法的可行性.数值结果表明,本文的模型在缓解交通拥挤方面可以得到更好的效果,而且只在部分路段进行路段能力扩充和收取电子路票更加方便实用.在可交易电子路票方案中,更多出行的用户需要购买电子路票来为他们的额外出行付费,而减少出行的用户则可以卖出多余电子路票得到补偿,同时电子路票的交易价格是在完全竞争的市场上形成的,因此本文中的可交易电子路票机制是收入中性的.     

A practical greedy approximation for the directed Steiner tree problem

The directed Steiner tree (DST) NP-hard problem asks, considering a directed weighted graph with n nodes and m arcs, a node r called root and a set of k nodes X called terminals, for a minimum cost directed tree rooted at r spanning X. The best known polynomial approximation ratio for DST is a \(O(k^\varepsilon )\)-approximation greedy algorithm. However, a much faster k-approximation, returning the shortest paths from r to X, is generally used in practice. We give two new algorithms : a fast k-approximation called Greedy\(_\text {FLAC}\) running in \(O(m \log (n)k + \min (m, nk)nk^2)\) and a \(O(\sqrt{k})\)-approximation called Greedy\(_\text {FLAC}^\triangleright \) running in \(O(nm + n^2 \log (n)k +n^2 k^3)\). We provide computational results to show that, Greedy\(_\text {FLAC}\) rivals in practice with the running time of the fast k-approximation and returns solution with smaller cost in practice.     

An approximation algorithm for the balanced Max-3-Uncut problem using complex semidefinite programming rounding

Graph partition problems have been investigated extensively in combinatorial optimization. In this work, we consider an important graph partition problem which has applications in the design of VLSI circuits, namely, the balanced Max-3-Uncut problem. We formulate the problem as a discrete linear program with complex variables and propose an approximation algorithm with an approximation ratio of 0.3456 using a semidefinite programming rounding technique along with a greedy swapping step afterwards to guarantee the balanced constraint. Our analysis utilizes a bivariate function, rather than the univariate function in previous work.     

A sustainable second-generation biodiesel supply chain network design problem under risk

This paper presents a multi-objective possibilistic programming model to design a second-generation biodiesel supply chain network under risk. The proposed model minimizes the total costs of biodiesel supply chain from feedstock supply centers to customer centers besides minimizing the environmental impact (EI) of all involved processes under a well-to-wheel perspective. Non-edible feedstocks are considered for biodiesel production. Variable cultivation cost of non-edible feedstock is assumed to be non-linear and dependent upon the amount of cultivated area. New formulation of possibilistic programming method is developed which is able to minimize the total mean and risk values of problems with possibilistic-based uncertainty. To solve the proposed multi-objective model, a hybrid solution approach based on flexible lexicographic and augmented ɛ-constraint methods is proposed which is capable to find appropriate efficient solutions from the Pareto-optimal set. The performance of the proposed possibilistic programming method as well as the developed solution approach are evaluated and validated through conducting a real case study in Iran. The outcome of this study demonstrates that high investment cost is required for improving the environmental impact and risk of sustainable biodiesel supply chain network design under risk. Decision maker preferences are required for suitable trade-off among total costs, risk values and environmental impact.     

A simplified algorithm for the depot location problem

This paper describes a simplified optimization algorithm used for the solution of a classical depot location problem as presented in a Greek Manufacturing Company. Algorithms in the literature for this type of problem are based on the assumption of predetermined fixed costs which are independent of the final size of the depots. This assumption is usually far from reality; the size of each depot does not remain constant during the optimization process and so does the associated fixed cost which is variable with the size of the depot. This assumption is relaxed in the proposed algorithm; the associated fixed cost is modified each time a new customer is allocated to a depot thus changing the required depot size.     

A memetic algorithm for the patient transportation problem

This paper addresses a real-life public patient transportation problem derived from the Hong Kong Hospital Authority (HKHA), which provides ambulance transportation services for disabled and elderly patients from one location to another. We model the problem as a multi-trip dial-a-ride problem (MTDARP), which requires designing several routes for each ambulance. A route is a sequence of locations, starting and terminating at the depot (hospital), according to which the ambulance picks up clients at the origins and delivers them to the destinations. A route is feasible only if it satisfies a series of side constraints, such as the pair and precedence constraints, capacity limit, ride time, route duration limit and time windows. Owing to the route duration limit, in particular, every ambulance is scheduled to operate several routes during the working period. To prevent the spread of disease, the interior of the ambulances needs to be disinfected at the depot between two consecutive trips. The primary aim of the problem investigated herein is to service more requests with the given resources, and to minimize the total travel cost for the same number of requests. In this paper, we provide a mathematical formulation for the problem and develop a memetic algorithm with a customized recombination operator. Moreover, the segment-based evaluation method is adapted to examine the moves quickly. The performance of the proposed algorithm is assessed using the real-world data from 2009 and compared with results obtained by solving the mathematical model. In addition, the proposed algorithm is adapted to solve the classic DARP instances, and found to perform well on medium-scale instances.     

A new effective branch-and-bound algorithm to the high order MIMO detection problem

This paper develops a branch-and-bound method based on a new convex reformulation to solve the high order MIMO detection problem. First, we transform the original problem into a \(\{-1,1\}\) constrained quadratic programming problem with the smallest size. The size of the reformulated problem is smaller than those problems derived by some traditional transformation methods. Then, we propose a new convex reformulation which gets the maximized average objective value as the lower bound estimator in the branch-and-bound scheme. This estimator balances very well between effectiveness and computational cost. Thus, the branch-and-bound algorithm achieves a high total efficiency. Several simulations are used to compare the performances of our method and other benchmark methods. The results show that this proposed algorithm is very competitive for high accuracy and relatively good efficiency.