Packing circles in the smallest circle: an adaptive hybrid algorithm

The circular packing problem (CPP) consists of packing n circles C _i of known radii r _i , i∈N={1,?…,?n}, into the smallest containing circle ?. The objective is to determine the coordinates (x _i ,?y _i ) of the centre of C _i , i∈N, as well as the radius r and centre (x,?y) of ?. CPP, which is a variant of the two-dimensional open-dimension problem, is NP hard. This paper presents an adaptive algorithm that incorporates nested partitioning within a tabu search and applies some diversification strategies to obtain a (near) global optimum. The tabu search is to identify the n circles’ ordering, whereas the nested partitioning is to determine the n circles’ positions that yield the smallest r. The computational results show the efficiency of the proposed algorithm.     

On the asymptotic optimality of a solution of the euclidean problem of covering a graph by <Emphasis Type="Italic">m</Emphasis> nonadjacent cycles of maximum total weight

In the problem of covering an n-vertex graph by m cycles of maximum total weight, it is required to find a family of m vertex-nonadjacent cycles such that it covers all vertices of the graph and the total weight of edges in the cover is maximum. The paper presents an algorithm for approximately solving the problem of covering a graph in Euclidean d-space R^d by m nonadjacent cycles of maximum total weight. The algorithm has time complexity O(n³). An estimate of the accuracy of the algorithm depending on the parameters d, m, and n is substantiated; it is shown that if the dimension d of the space is fixed and the number of covering cycles is m = o(n), then the algorithm is asymptotically exact.     

The pos/neg-weighted 2-medians in balanced trees with subtree-shaped customers

This paper deals with the pos/neg-weighted p-median problem on tree graphs where all customers are modeled as subtrees. We present a polynomial algorithm for the 2-median problem on an arbitrary tree. Then we improve the time complexity to O(n log n) for the problem on a balanced tree, where n is the number of the vertices in the tree.     

A note on scheduling jobs with equal processing times and inclusive processing set restrictions

We consider the problem of scheduling n jobs on m parallel machines with inclusive processing set restrictions. Each job has a given release date, and all jobs have equal processing times. The objective is to minimize the makespan of the schedule. Li and Li (2015) have developed an O(n²+mn log?n) time algorithm for this problem. In this note, we present a modified algorithm with an improved time complexity of O(min{m, log?n} ? n log?n).     

A continuation method for solving symmetric Toeplitz systems

A fast algorithm is proposed for solving symmetric Toeplitz systems. This algorithm continuously transforms the identity matrix into the inverse of a given Toeplitz matrix T. The memory requirements for the algorithm are O(n), and its complexity is O(log κ(T)nlogn), where (T) is the condition number of T. Numerical results are presented that confirm the efficiency of the proposed algorithm.     

Multi-period empty container repositioning with stochastic demand and lost sales

This paper considers repositioning empty containers between multi-ports over multi-periods with stochastic demand and lost sales. The objective is to minimize the total operating cost including container-holding cost, stockout cost, importing cost and exporting cost. First, we formulate the single-port case as an inventory problem over a finite horizon with stochastic import and export of empty containers. The optimal policy for period n is characterized by a pair of critical points (A _n , S _n ), that is, importing empty containers up to A _n when the number of empty containers in the port is fewer than A _n ; exporting empty containers down to S _n when the number of empty containers in the port is more than S _n ; and doing nothing, otherwise. A polynomial-time algorithm is developed to determine the two thresholds, that is, A _n and S _n , for each period. Next, we formulate the multi-port problem and determine a tight lower bound on the cost function. On the basis of the two-threshold optimal policy for a single port, a polynomial-time algorithm is developed to find an approximate repositioning policy for multi-ports. Simulation results show that the proposed approximate repositioning algorithm performs very effectively and efficiently.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

Scheduling Independent Tasks with Deadlines on Semi-identical Processors

Given m semi-identical processors which are parallel processors all working with the same speed but in different time intervals of availability and n independent tasks with deadlines, the problem of constructing a feasible pre-emptive schedule is examined. We present an O (nm log n) time algorithm to construct such a schedule whenever one exists. We show that the number of induced pre-emptions is proportional to the total number of processing intervals and deadlines.     

Iterated local search with Trellis-neighborhood for the partial Latin square extension problem

A partial Latin square (PLS) is a partial assignment of n symbols to an \(n\times n\) grid such that, in each row and in each column, each symbol appears at most once. The PLS extension problem is an NP-hard problem that asks for a largest extension of a given PLS. We consider the local search such that the neighborhood is defined by (p, q)-swap , i.e., the operation of dropping exactly p symbols and then assigning symbols to at most q empty cells. As a fundamental result, we provide an efficient \((p,\infty )\)-neighborhood search algorithm that finds an improved solution or concludes that no such solution exists for \(p\in \{1,2,3\}\). The running time of the algorithm is \(O(n^{p+1})\). We then propose a novel swap operation, Trellis-swap, which is a generalization of (p, q)-swap with \(p\le 2\). The proposed Trellis-neighborhood search algorithm runs in \(O(n^{3.5})\) time. The iterated local search (ILS) algorithm with Trellis-neighborhood is more likely to deliver a high-quality solution than not only ILSs with \((p,\infty )\)-neighborhood but also state-of-the-art optimization solvers such as IBM ILOG CPLEX and LocalSolver.     

An exact algorithm for finding a vector subset with the longest sum

We consider the problem: Given a set of n vectors in the d-dimensional Euclidean space, find a subsetmaximizing the length of the sum vector.We propose an algorithm that finds an optimal solution to this problem in time O(n^d?1(d + logn)). In particular, if the input vectors lie in a plane then the problem is solvable in almost linear time.     

Guided construction search metaheuristics for the capacitated p-median problem with single source constraint

In the capacitated p-median problem with single source constraint, also known as the capacitated clustering problem, a given set of n weighted points is to be partitioned into p clusters such that the total weight of the points in each cluster does not exceed a given cluster capacity. The objective is to find a set of p centres that minimizes the total scatter of points allocated to these clusters. In this paper, a (λ, μ)-interchange neighbourhood based on the concept of λ-interchange of points restricted to μ-adjacent clusters is proposed. Structural properties of centres are identified and exploited to derive special data structures for their efficient evaluations. Different search and selection strategies including the variable neighbourhood search descent with respect to μ-nearest points are investigated. The most efficient strategies are then embedded in a guided construction search metaheuristic framework based either on a periodic local search procedure or a greedy random adaptive search procedure to solve the problem. Computational experience is reported on a standard set of benchmarks. The computational experience demonstrates the competitive performance of the proposed algorithms when compared to the best-known procedures in the literature in terms of solution quality and computational requirement.     

An approximation algorithm for finding a <Emphasis Type="Italic">d</Emphasis>-regular spanning connected subgraph of maximum weight in a complete graph with random weights of edges

An approximation algorithm is suggested for the problem of finding a d-regular spanning connected subgraph of maximum weight in a complete undirected weighted n-vertex graph. Probabilistic analysis of the algorithm is carried out for the problem with random input data (some weights of edges) in the case of a uniform distribution of the weights of edges and in the case of a minorized type distribution. It is shown that the algorithm finds an asymptotically optimal solution with time complexity O(n ²) when d = o(n). For the minimization version of the problem, an additional restriction on the dispersion of weights of the graph edges is added to the condition of the asymptotical optimality of the modified algorithm.     

Two-stage flexible flow shop scheduling subject to fixed job sequences

This paper investigates the scheduling problem in a two-stage flexible flow shop, which consists of m stage-1 parallel dedicated machines and a stage-2 bottleneck machine, subject to the condition that n _l jobs per type l∈{1, …, m} are processed in a fixed sequence. Four regular performance metrics, including the total completion time, the maximum lateness, the total tardiness, and the number of tardy jobs, are considered. For each considered objective function, we aim to determine an optimal interleaving processing sequence of all jobs coupled with their starting times on the stage-2 bottleneck machine. The problem under study is proved to be strongly NP-hard. An O(m²Π_l=1 ^m n _l ²) dynamic programming algorithm coupled with numerical experiments is presented.     

Two-stage assembly-type flowshop batch scheduling problem subject to a fixed job sequence

This paper discusses a two-stage assembly-type flowshop scheduling problem with batching considerations subject to a fixed job sequence. The two-stage assembly flowshop consists of m stage-1 parallel dedicated machines and a stage-2 assembly machine which processes the jobs in batches. Four regular performance metrics, namely, the total completion time, maximum lateness, total tardiness, and number of tardy jobs, are considered. The goal is to obtain an optimal batching decision for the predetermined job sequence at stage 2. This study presents a two-phase algorithm, which is developed by coupling a problem-transformation procedure with a dynamic program. The running time of the proposed algorithm is O(mn+n⁵), where n is the number of jobs.     

Balancing Workloads and Minimizing Set-up Costs in the Parallel Processing Shop

This paper presents an optimal scheduling algorithm for minimizing set-up costs in the parallel processing shop while meeting workload balancing restrictions.There are M independent batch type jobs which have sequence dependent set-up costs and N parallel processing machines. Each of the M jobs must be processed on exactly one of the N available machines. It is desirable to minimize total changeover costs with the restriction that each machine workload assignment T _n be within P units of the average machine assignment. The paper describes a static problem in which all jobs are available at time zero. The sequence dependent change over costs are identical for each machine. An extension of the algorithm handles nonidentical processor problems.A combinatorial programming approach to the problem is used. For the special case of identical processors, the problem can be treated as a multi-salesman travelling salesman problem. A general branch and bound algorithm and numerical results are given.     

Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles

We consider the m-Cycle Cover Problem of covering a complete undirected graph by m vertex-nonadjacent cycles of extremal total edge weight. The so-called TSP approach to the construction of an approximation algorithm for this problem with the use of a solution of the traveling salesman problem (TSP) is presented. Modifications of the algorithm for the Euclidean Max m-Cycle Cover Problem with deterministic instances (edge weights) in a multidimensional Euclidean space and the Random Min m-Cycle Cover Problem with random instances UNI(0,1) are analyzed. It is shown that both algorithms have time complexity O(n ³) and are asymptotically optimal for the number of covering cycles m = o(n) and \(m \leqslant \frac{{n^{1/3} }}{{\ln n}}\), respectively.     

A Polynomial Time Algorithm for the Resource Allocation Problem with a Convex Objective Function

Consider the resource allocation problem:minimize ∑ⁿ_i=1 f_i(x_i) subject to ∑ⁿ_i=1 x_i = N and x_i's being nonnegative integers, where each f_i is a convex function. The well-known algorithm based on the incremental method requires O(N log n + n) time to solve this problem. We propose here a new algorithm based on the Lagrange multiplier method, requiring O[n²(log N)²] time. The latter is faster if N is much larger than n. Such a situation occurs, for example, when the optimal sample size problem related to monitoring the urban air pollution is treated.     

Radar placement along banks of river

In this paper, we consider the Radar Placement and Power Assignment problem (RPPA) along a river. In this problem, a set of crucial points in the river are required to be monitored by a set of radars which are placed along the two banks. The goal is to choose the locations for the radars and assign powers to them such that all the crucial points are monitored and the total power is minimized. If each crucial point is required to be monitored by at least k radars, the problem is a k-Coverage RPPA problem (k-CRPPA). Under the assumption that the river is sufficiently smooth, one may focus on the RPPA problem along a strip (RPPAS). In this paper, we present an O(n ⁹) dynamic programming algorithm for the RPPAS, where n is the number of crucial points to be monitored. In the special case where radars are placed only along the upper bank, we present an O(kn ⁵) dynamic programming algorithm for the k-CRPPAS. For the special case that the power is linearly dependent on the radius, we present an O(n log n)-time \({2\sqrt 2}\)-approximation algorithm for the RPPAS.     

Solution of the NP-hard total tardiness minimization problem in scheduling theory

The classical NP-hard (in the ordinary sense) problem of scheduling jobs in order to minimize the total tardiness for a single machine 1‖ΣT _j is considered. An NP-hard instance of the problem is completely analyzed. A procedure for partitioning the initial set of jobs into subsets is proposed. Algorithms are constructed for finding an optimal schedule depending on the number of subsets. The complexity of the algorithms is O(n ²Σp _j ), where n is the number of jobs and p _j is the processing time of the jth job (j = 1, 2, …, n).     

The connected <Emphasis Type="Italic">p</Emphasis>-median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.