A heuristic for the Stacker Crane Problem on trees which is almost surely exact

Given an edge-weighted transportation network G and a list of transportation requests L, the Stacker Crane Problem is to find a minimum-cost tour for a server along the edges of G that serves all requests. The server has capacity one, and starts and stops at the same vertex. In this paper, we consider the case that the transportation network G is a tree, and that the requests are chosen randomly according to a certain class of probability distributions. We show that a polynomial time algorithm by Frederickson and Guan [J. Algorithms 15 (1993) 29-60], which guarantees a 4/3-approximation in the worst case, on almost all inputs finds a minimum-cost tour, along with a certificate of the optimality of its output.

[1]  René Beier,et al.  Random knapsack in expected polynomial time , 2003, STOC '03.

[2]  Colin McDiarmid,et al.  Algorithmic theory of random graphs , 1997 .

[3]  Martin Grötschel,et al.  Online optimization of large scale systems , 2001 .

[4]  Chul E. Kim,et al.  Approximation Algorithms for Some Routing Problems , 1978, SIAM J. Comput..

[5]  Sven Oliver Krumke,et al.  Online Dial-a-Ride Problems: Minimizing the Completion Time , 2000, STACS.

[6]  S. Hougardy,et al.  Approximation Algorithms for the Steiner Tree Problem in Graphs , 2001 .

[7]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[8]  Santosh S. Vempala,et al.  On The Approximability Of The Traveling Salesman Problem , 2006, Comb..

[9]  Greg N. Frederickson,et al.  Nonpreemptive Ensemble Motion Planning on a Tree , 1993, J. Algorithms.

[10]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[11]  Donald E. Knuth,et al.  The Art of Computer Programming: Volume 3: Sorting and Searching , 1998 .

[12]  Donald Ervin Knuth,et al.  The Art of Computer Programming, 2nd Ed. (Addison-Wesley Series in Computer Science and Information , 1978 .

[13]  Moses Charikar,et al.  The finite capacity dial-a-ride problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[14]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[15]  D. Du,et al.  Steiner Trees in Industry , 2002 .

[16]  Leen Stougie,et al.  Algorithms for the On-Line Travelling Salesman1 , 2001, Algorithmica.

[17]  Alan M. Frieze.  On random symmetric travelling salesman problems , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[18]  Dih Jiun Guan,et al.  Routing a Vehicle of Capacity Greater than one , 1998, Discret. Appl. Math..

[19]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[20]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[21]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[22]  Mikhail J. Atallah,et al.  Efficient Solutions to Some Transportation Problems with Applications to Minimizing Robot Arm Travel , 1988, SIAM J. Comput..

[23]  Santosh S. Vempala,et al.  On The Approximability Of The Traveling Salesman Problem , 2006, Comb..

[24]  Till Nierhoff,et al.  Greedily Approximating the r-independent Set and k-center Problems on Random Instances , 1997, RANDOM.

[25]  Sven Oliver Krumke,et al.  Euler is Standing in Line , 1999, WG.

[26]  Donald E. Knuth,et al.  Sorting and Searching , 1973 .

[27]  Sven Oliver Krumke,et al.  Euler is standing in line dial-a-ride problems with precedence-constraints , 2001, Discret. Appl. Math..

[28]  Günter Rote,et al.  Vehicle routing in an automated warehouse: Analysis and optimization , 1995, Ann. Oper. Res..

[29]  Leen Stougie,et al.  On-line single-server dial-a-ride problems , 2001, Theor. Comput. Sci..

[30]  Leen Stougie,et al.  Competitive Algorithms for the On-line Traveling Salesman , 1995, WADS.

[31]  Ravi B. Boppana,et al.  Eigenvalues and graph bisection: An average-case analysis , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).