combinatorial optimization Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combina ...

, the stacker crane problem is an optimization problem closely related to the traveling salesperson problem. Its input consists of a collection of

ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...

s of points in a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

, and the goal is to connect these points into a cycle of minimum total length that includes all of the pairs, oriented consistently with each other. It models problems of scheduling the pickup and delivery of individual loads of cargo, by a stacker crane, construction crane or (in

drayage Drayage is the transportation of shipping containers by truck to its final destination. Drayage is often part of a longer overall move, such as from a ship to a warehouse. Some research defines it specifically as "a truck pickup from or delive ...

) a truck, in a simplified form without constraints on the timing of these deliveries. It was introduced by , with an equivalent formulation in terms of

mixed graph In graph theory, a mixed graph is a graph consisting of a set of vertices , a set of (undirected) edges , and a set of directed edges (or arcs) . Definitions and notation Consider adjacent vertices u,v \in V. A directed edge, called an arc, ...

s with directed edges modeling the input pairs and undirected edges modeling their distances. Frederickson et al. credit its formulation to a personal communication of Daniel J. Rosenkrantz. The stacker crane problem can be viewed as a generalization of the traveling salesperson problem in metric spaces: any instance of the traveling salesperson problem can be transformed into an instance of the stacker crane problem, having a pair

(p,p)

for each point in the travelling salesman instance. In the other direction, the stacker crane problem can be viewed as a special case of the asymmetric traveling salesperson problem, where the points of the asymmetric traveling salesperson problem are the pairs of a stacker crane instance and the distance from one pair to another is taken as the distance from the delivery point of the first pair, through its pickup point, to the delivery point of the second pair. Because it generalizes the traveling salesperson problem, it inherits the same

computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...

: it is

NP-hard In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assumi ...

, and at least as hard to approximate. An

approximation algorithm In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned sol ...

based on the Christofides algorithm for the traveling salesperson problem can approximate the solution of the stacker crane problem to within an approximation ratio of 9/5. The problem of designing the back side of an

embroidery Embroidery is the art of decorating Textile, fabric or other materials using a Sewing needle, needle to stitch Yarn, thread or yarn. It is one of the oldest forms of Textile arts, textile art, with origins dating back thousands of years across ...

pattern to minimize the total amount of thread used is closely related to the stacker crane problem, but it allows each of its pairs of points (the ends of the visible stitches on the front side of the pattern) to be traversed in either direction, rather than requiring the traversal to go through all pairs in a consistent direction. It is NP-hard by the same transformation from the traveling salesperson problem, and can be approximated to within an approximation ratio of 2. Another variation of the stacker crane problem, called the dial-a-ride problem, asks for the minimum route for a vehicle to perform a collection of pickups and deliveries while allowing it to hold some number of loads at any point along its route.

References

{{reflist, refs= {{citation , last1 = Arkin , first1 = Esther M. , author1-link = Esther Arkin , last2 = Hart , first2 = George , last3 = Kim , first3 = Joondong , last4 = Kostitsyna , first4 = Irina , last5 = Mitchell , first5 = Joseph S. B. , author5-link = Joseph S. B. Mitchell , last6 = Sabhnani , first6 = Girishkumar , last7 = Skiena , first7 = Steven , author7-link = Steven Skiena , contribution = The Embroidery Problem , title = Proceedings of the 20th Annual Canadian Conference on Computational Geometry, Montréal, Canada, August 13-15, 2008 , year = 2008 {{citation , last1 = Charikar , first1 = Moses , author1-link = Moses Charikar , last2 = Raghavachari , first2 = Balaji , contribution = The Finite Capacity Dial-A-Ride Problem , doi = 10.1109/SFCS.1998.743496 , pages = 458–467 , publisher = IEEE Computer Society , title = 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA , year = 1998, isbn = 0-8186-9172-7 {{citation , last1 = Frederickson , first1 = Greg N. , last2 = Hecht , first2 = Matthew S. , last3 = Kim , first3 = Chul E. , doi = 10.1137/0207017 , issue = 2 , journal =

SIAM Journal on Computing The ''SIAM Journal on Computing'' is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM). Although its official ISO abbreviation i ...

, mr = 489787 , pages = 178–193 , title = Approximation algorithms for some routing problems , volume = 7 , year = 1978; previously announced at the 17th Annual Sympoisum on Foundations of Computer Science, 1976 {{citation, last=Srour, first=F. J., year=2010, title=Dissecting Drayage: An Examination of Structure, Information, and Control in Drayage Operations, volume=EPS-2010-186-LIS, series=ERIM Ph.D. Series Research in Management, publisher=Erasmus Research Institute of Management, hdl=1765/18231, isbn=978-90-5892-226-7 Travelling salesman problem