Three-index Assignment Problem Research Articles

Solution approaches to the three-index assignment problem

This paper delves into the axial Three-index Assignment Problem (3IAP), alternatively known as the Multi-dimensional Assignment Problem, defined as an extension of the classical two-dimensional assignment problem. The 3IAP entails allocating $n$ tasks to $n$ machines in $n$ factories, ensuring one task is completed by one machine in one factory at a minimum total cost. This combinatorial optimization problem is classified as \emph{NP}-hard due to its inherent complexity and being the subject of much scholarly research and investigation. The study employs an algorithmic approach to devise rapid and effective solutions for the 3IAP. A new heuristic Greedy-style Procedure (GSP) is introduced for solving the 3IAP, achieving feasible solutions within polynomial time. Particular configurations of cost matrices enable us to reach quality solutions. Examining tie-cases and matrix ordering unveiled innovative variants. Further investigation of cost matrix attributes facilitates the development of two new heuristic categories, offering optimal or nearly optimal solutions for the 3IAP. Extensive numerical experiments validate the effectiveness of the heuristics, generating quality solutions in a short computational time. Furthermore, we implement two potent methods using optimization solvers, achieving optimal solutions for the 3IAP within competitive CPU times.

An Adaptive Algorithm for Solving the Axial Three-Index Assignment Problem

In this paper, a probabilistic modification of the minimal element algorithm for solving the axial three-index assignment problem is suggested. Its general idea is to extend the basic greedy-type algorithmic schemes using transition to a probabilistic setup based on variables randomization. The minimization of an objective function is replaced by the minimization of its expectation. The algorithm is implemented in three stages as follows. At the first stage, a motion in the set of random variables is defined. At the second stage, an inequality that expresses the local improvement condition is solved. At the third stage, the probabilities are recalculated, which represents an “adaptation” process. The second stage reveals a feature of the algorithm: the resulting solution depends on the “qualities” of the element itself and also on possible losses of its choice.

On a family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation

В 1995 году Т. Мацуи рассмотрел специальное семейство 0/1-многогранников с NP-полным критерием несмежности вершин. В 2012 году автор настоящей работы установил, что все многогранники этого семейства присутствуют в качестве граней в многогранниках, ассоциированных со следующими NP-полными задачами: задачей коммивояжера, задачей 3-выполнимость, задачей о рюкзаке, задачей о покрытии множества, задачей о частичном упорядочивании, задачей о кубическом подграфе и некоторыми другими. В работе показано, что ни один из многогранников упомянутого выше специального семейства, за исключением одномерного отрезка, не может быть гранью многогранников, ассоциированных с задачами о независимом множестве в графе, об упаковке и разбиении множества и с трехиндексной задачей о назначениях.

Fast separation for the three-index assignment problem

A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the number of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x, denoted as supp(x), which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x, we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k-index assignment problem and we show empirically how the separation algorithms exploiting sparsity improve on existing ones by running them on the largest instances reported in the literature.

Efficient algorithms with performance guarantees for some problems of finding several discrete disjoint subgraphs in complete weighted graph

Some hard-to solve combinatorial problems of finding several disjoint discrete structures in complete weighted graph are considered. Efficient algorithms with performance guarantees are constructed for the Euclidean m-Peripatetic Salesman Problem, m-Weighted Clique Problem and m-Layer Planar three-index Assignment Problem.

The Focus of Attention Problem

We consider systems of small, cheap, simple sensors that are organized in a distributed network and used for estimating and tracking the locations of targets. The objective is to assign sensors to targets such that the overall expected error of the sensors’ estimates of the target locations is minimized. The so-called focus of attention problem (FOA) deals with the special case where every target is tracked by one pair of range sensors. The resulting computational problem is a special case of the axial three-index assignment problem, a well-known fundamental problem in combinatorial optimization. We provide a complete complexity and approximability analysis of the FOA problem: we establish strong NP-hardness and the unlikeliness of an FPTAS, we identify polynomially solvable special cases, and we construct a sophisticated polynomial time approximation scheme for it.

On the Complexity of Separation: The Three-Index Assignment Problem

A fundamental step in any cutting plane algorithm is separation: deciding whether a violated inequality exists within a certain class of inequalities. It is customary to express the complexity of a separation algorithm in n, the number of variables. Here, we argue that the input to a separation algorithm can be expressed in jsup(x)j, where sup(x) denotes the vector containing the positive components of x. This input measure allows one to take sparsity into account. We apply this idea to two known classes of valid inequalities for the three-index assignment problem, and we find separation algorithms with a better complexity than the ones known in literature. We also show empirically the performance of our separation algorithms.

Between a rock and a hard place: the two-to-one assignment problem

We describe the two-to-one assignment problem, a problem in between the axial three-index assignment problem and the three-dimensional matching problem, having applications in various domains. For the (relevant) case of decomposable costs satisfying the triangle inequality we provide, on the positive side, two constant factor approximation algorithms. These algorithms involve solving minimum weight matching problems and transportation problems, leading to a 2-approximation, and a $${\frac{3}{2}}$$ -approximation. Moreover, we further show that the best of these two solutions is a $${\frac{4}{3}}$$ -approximation for our problem. On the negative side, we show that the existence of a polynomial time approximation scheme for our problem would imply P = NP. Finally, we report on some computational experiments showing the performance of the described heuristics.

On the Computational Complexity of Peer-to-Peer Satellite Refueling Strategies

We revisit the peer-to-peer on-orbit satellite refueling problem, in which the maneuvering satellites are allowed to interchange their orbital positions. We show that the problem is computationally hard by reducing it to a special case of the three-index assignment problem. On the positive side, we show that the size of instances in practice is such that a state-of-the-art integer programming solver is able to find globally optimal solutions in little computing time.

Lower bounds for the axial three-index assignment problem

This paper describes new bounding methods for the axial three-index assignment problem (3AP). For calculating 3AP lower bounds, we use a projection method followed by a Hungarian algorithm, based on a new Lagrangian relaxation. We also use a cost transformation scheme, which iteratively transforms 3AP costs in a series of equivalent 3APs, which provides the possibility of improving the 3AP lower bound. These methods produce efficiently computed relatively tight lower bound.

Round robin tournaments and three index assignments

Scheduling a sports league can be seen as a difficult combinatorial optimization problem. We study some variants of round robin tournaments and analyze the relationship with the planar three-index assignment problem. The complexity of scheduling a minimum cost round robin tournament is established by a reduction from the planar three-index assignment problem. Furthermore, we introduce integer programming models. We pick up a popular idea and decompose the overall problem in order to obtain two subproblems which can be solved sequentially. We show that the latter subproblem can be casted as a planar three-index assignment problem. This makes existing solution techniques for the planar three-index assignment problem amenable to sports league scheduling.

A Note on Peer-to-Peer Satellite Refueling Strategies

We revisit the peer-to-peer refueling problem, in which the maneuvering satellites are allowed to interchange their orbital positions [2]. We show that the problem is computationally hard, by reducing it to a special case of the three-index assignment problem. On the positive side, we show that the size of instances from practice is such that a state-of-the-art integer programming solver is able to find optimal solutions in little computing time.

An asymptotically exact algorithm for one modification of planar three-index assignment problem

An m-layer three-index assignment problem is considered which is a modification of the classical planar three-index assignment problem. This problem is NP-hard for m ⩾ 2. An approximate algorithm, solving this problem for 1 an > 0. The time complexity of the algorithm is O(mn2 + m7/2). It is shown that in the case of a uniform distribution (and also a distribution of minorized type) the algorithm is asymptotically exact if m = Θ(n1 − θ) and bn/an = o(nθ) for every constant θ, 0 < θ < 1.

A hybrid genetic algorithm for the Three-Index Assignment Problem

The Three-Index Assignment Problem (AP3) is well-known problem which has been shown to be NP -hard. This problem has been studied extensively, and many exact and heuristic methods have been proposed to solve it. Inspired by the classical assignment problem, we propose a new local search heuristic which solves the problem by simplifying it to the classical assignment problem. We further hybridize our heuristic with the genetic algorithm (GA). Extensive experimental results indicate that our hybrid method is superior to all previous heuristic methods including those proposed by Balas and Saltzman [Operations Research 39 (1991) 150–161], Crama and Spieksma [European Journal of Operational Research 60 (1992) 273–279], Burkard et al. [Discrete Applied Mathematics 65 (1996) 123–169], and Aiex et al. [GRASP with path relinking for the three-index assignment problem, Technical report, INFORMS Journal on Computing, in press. Available from: <http://www.research.att.com/~mgcr/doc/g3index.pdf>].

GRASP with Path Relinking for Three-Index Assignment

This paper proposes and tests variants of GRASP (greedy randomized adaptive search procedure) with path relinking for the three-index assignment problem (AP3). GRASP is a multistart metaheuristic for combinatorial optimization. It usually consists of a construction procedure based on a greedy randomized algorithm and of a local search. Path relinking is an intensification strategy that explores trajectories that connect high-quality solutions. Several variants of the heuristic are proposed and tested. Computational results show clearly that this GRASP for AP3 benefits from path relinking and that the variants considered in this paper compare well with previously proposed heuristics for this problem. GRASP with path relinking was able to improve the solution quality of heuristics proposed by Balas and Saltzman (1991), Burkard et al. (1996), and Crama and Spieksma (1992) on all instances proposed in those papers. We show that the random variable “time to target solution,” for all proposed GRASP with path-relinking variants, fits a two-parameter exponential distribution. To illustrate the consequence of this, one of the variants of GRASP with path relinking is shown to benefit from parallelization.

Spectral decompositions and feasible directions in the axial three-index assigment problem

In this paper, we give some results on the spectral deconposition and generalized inverse of the matrİK A whidı is the coeffident maJrix of the axial three-index assignment problem and investigate relations between eigenvalues and eigenvectors of the matrices and I-A' A where A’^ is the transpoze and A^ is the generalized inverse of A. It has been shown that the feasible directions of the axial three- index assignment problem can be investigated in terms ofthe eigenvectors oft he matrix AA7

Tabu Search for the planar three-index assignment problem

Tabu Search is a very effective method for approximately solving hard combinatorial problems. In the current work we implement Tabu Search for solving the Planar Three-Index Assignment Problem. The problem deals with finding the minimum cost assignment between elements of three distinct sets demanding that every pair of elements, each representing a different set, appears in the same solution exactly once. The solutions of the problems correspond to Latin squares. These structures form the basis of the move generation mechanism employed by the Tabu Search procedures. Standard Tabu Search ideas such as candidate move list, variable tabu list size, and frequency-based memory are tested. Computational results for a range of problems of varying dimensions are presented.

An algorithm for the planar three-index assignment problem

A branch and bound algorithm for the planar three-index assignment problem is presented. The branching operation is performed on sets of variables belonging to a common constraint rather than on a single variable. The algorithm comprises an upper and lower bound procedure. The upper bound procedure consists of two parts: one for finding an initial solution and one for improving it. The local improvement rule takes advantage of the Latin square structure of the solution. The lower bound procedure combines a new dual heuristic and a Lagrangean relaxation scheme solved by subgradient optimization. The use of reduced costs in the evaluation of bounds is another feature of the algorithm. Computational experience both on the exact algorithm and on the primal heuristics as standalone procedures is reported.

Linear-time separation algorithms for the three-index assignment polytope

Balas and Saltzman identified several classes of facet inducing inequalities for the three-index assignment polytope, and gave O( n 4 ) separation algorithms for two of them. We give O( n 3 ) separation algorithms for these two classes of facets, and also for a third class. Since the three-index assignment problem has n 3 variables, these algorithms are linear-time and their complexity is best possible.

An Algorithm for the Three-Index Assignment Problem

We describe a branch-and-bound algorithm for solving the axial three-index assignment problem. The main features of the algorithm include a Lagrangian relaxation that incorporates a class of facet inequalities and is solved by a modified subgradient procedure to find good lower bounds, a primal heuristic based on the principle of minimizing maximum regret plus a variable depth interchange phase for finding good upper bounds, and a novel branching strategy that exploits problem structure to fix several variables at each node and reduce the size of the total enumeration tree. Computational experience is reported on problems with up to 78 equations and 17,576 variables. The primal heuristics were tested on problems with up to 210 equations and 343,000 variables.