Limited column generation and related methods for local access telecommunication network design and expansion-formation, algorithm, and implementation

Geon Cho, Purdue University

Abstract

This thesis presents models, algorithms and implementation techniques for Local Access Telecommunication Network (LATN) Design, Expansion and related problems. In particular, we develop detailed algorithms for implementing the so-called Limited Column Generation procedure. We formulate both problems into a tree-partitioning problem with an exponential number of variables. Its linear programming relaxation has all integral vertices, and can be solved by the Limited Column Generation procedure in just n pivots, where n is the number of nodes in the network. Prior to each pivot, an entering variable is selected by detecting the Locally Most Violated (LMV) reduced cost, which can be obtained by solving a subproblem in pseudo-polynomial time. The corresponding subproblems for the LATN Design and Expansion problem are called the Centered Tree Knapsack Problem (CTKP) and the Extended Tree Knapsack Problem (ETKP) with arc expansion cost, respectively. A critical step in the Limited Column Generation is to find all the LMV reduced costs. As dual variables are updated at each pivot, the reduced costs have to be computed in an on-line fashion. An efficient implementation is developed to execute such a task so that the LATN Design problem can be solved in $O(n\sp2H)$, and the LATN Expansion problem can be solved in $O(n\sp2\delta H)$ time, where $\delta(\leq n)$ is the depth of the tree and H is the maximum concentrator capacity. Our computational experiments indicate that our algorithm delivers an outstanding performance. For instance, the LATN Design and Expansion problems with 150 nodes can be solved in approximately 67 seconds and 340 seconds, respectively, on a SUN SPARC 1000 workstation. The Tree Knapsack Problem (TKP) can be regarded as a 0-1 Knapsack Problem on a rooted tree T subject to a node precedence relation, that is, if a node is included into the knapsack, then its predecessor must be also included. In this thesis, we develop both a branch-and-bound algorithm and a "depth-first" dynamic programming algorithm for solving the TKP. A TKP can be solved by our "depth-first" dynamic programming algorithm in O(nH) time.

Degree

Ph.D.

Advisors

Shaw, Purdue University.

Subject Area

Industrial engineering|Operations research|Mathematics

COinS