Steiner tree reoptimization in graphs with sharpened triangle inequality

In this paper, we deal with several reoptimization variants of the Steiner tree problem in graphs obeying a sharpened @b-triangle inequality. A reoptimization algorithm exploits the knowledge of an optimal solution to a problem instance for finding good solutions for a locally modified instance. We show that, in graphs satisfying a sharpened triangle inequality (and even in graphs where edge-costs are restricted to the values 1 and 1+@c for an arbitrary small @c>0), Steiner tree reoptimization still is NP-hard for several different types of local modifications, and even APX-hard for some of them. As for the upper bounds, for some local modifications, we design linear-time (1/2+@b)-approximation algorithms, and even polynomial-time approximation schemes, whereas for metric graphs (@b=1), none of these reoptimization variants is known to permit a PTAS. As a building block for some of these algorithms, we employ a 2@b-approximation algorithm for the classical Steiner tree problem on such instances, which might be of independent interest since it improves over the previously best known ratio for any @b<1/2+ln(3)/4~0.775.

[1]  Dennis Komm,et al.  Reoptimization of the metric deadline TSP , 2008, J. Discrete Algorithms.

[2]  F. Hwang,et al.  The Steiner Tree Problem , 2012 .

[3]  Dennis Komm,et al.  Reoptimization of the Shortest Common Superstring Problem , 2009, Algorithmica.

[4]  Alex Zelikovsky,et al.  Improved Steiner tree approximation in graphs , 2000, SODA '00.

[5]  Juraj Hromkovic,et al.  Reoptimization of Steiner trees: Changing the terminal set , 2009, Theor. Comput. Sci..

[6]  Vangelis Th. Paschos,et al.  Simple and Fast Reoptimizations for the Steiner Tree Problem , 2009, Algorithmic Oper. Res..

[7]  David Fernández-Baca,et al.  On the Approximability of the Steiner Tree Problem in Phylogeny , 1996, ISAAC.

[8]  Markus W. Schäffter,et al.  Scheduling with Forbidden Sets , 1997, Discret. Appl. Math..

[9]  Panos M. Pardalos,et al.  Steiner Tree Problems , 2009, Encyclopedia of Optimization.

[10]  Vangelis Th. Paschos,et al.  Reoptimization of minimum and maximum traveling salesman's tours , 2009, J. Discrete Algorithms.

[11]  David Fernández-Baca,et al.  On the Approximability of the Steiner Tree Problem in Phylogeny , 1996, Discret. Appl. Math..

[12]  Marshall W. Bern,et al.  The Steiner Problem with Edge Lengths 1 and 2 , 1989, Inf. Process. Lett..

[13]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[14]  Guido Proietti,et al.  Reusing Optimal TSP Solutions for Locally Modified Input Instances , 2006, IFIP TCS.

[15]  Anna Zych,et al.  Reoptimization of Weighted Graph and Covering Problems , 2008, WAOA.

[16]  Luca Bertazzi,et al.  Reoptimizing the 0-1 knapsack problem , 2010, Discret. Appl. Math..

[17]  Juraj Hromkovic,et al.  On the Hardness of Reoptimization , 2008, SOFSEM.

[18]  Luca Bertazzi,et al.  Reoptimizing the traveling salesman problem , 2003, Networks.

[19]  Hans Jürgen Prömel,et al.  The Steiner Tree Problem , 2002 .

[20]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

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

[22]  Juraj Hromkovic,et al.  Reoptimization of Steiner Trees , 2008, SWAT.

[23]  Yoji Kajitani,et al.  Approximating Steiner trees in graphs with restricted weights , 1998, Networks.

[24]  Juraj Hromkovic,et al.  Approximation algorithms for the TSP with sharpened triangle inequality , 2000, Inf. Process. Lett..