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An exact column generation algorithm for load balancing in capacity sharing networks

Kaixiang Hu, Feilong Huang, Caixia Kou

TL;DR

This work addresses load balancing in capacity sharing networks by formulating it as a maximum multi-commodity flow problem with per-technology capacity constraints. It introduces CGLAD, an exact column generation algorithm that converts the NP-hard optimality check into a single-constrained shortest path problem and solves it exactly using BiLAD and ExactBiLAD, guided by Lagrange duality. The method achieves optimal network throughput while complying with delay and technology-capacity constraints, and its performance is competitive with heuristic approaches while outperforming other exact methods on large instances. Numerical experiments on 200 randomly generated networks show CGLAD delivers strong throughput gains and strict delay adherence with computation times on par with state-of-the-art heuristics. Overall, CGLAD provides a robust benchmark and practical exactness for throughput optimization in capacity sharing networks.

Abstract

Capacity sharing networks are typical heterogeneous communication networks widely applied in information and communications technology (ICT) field. In such networks, resources like bandwidth, spectrum, computation and storage are shared among various communication services. Meanwhile, the issue of network congestion is always a prominent challenge. To handle network congestion essentially needs to solve the load balancing of networks. In this paper, for capacity sharing networks, we formulate their load balancing problem as a maximum multi-commodity flow problem. For such a problem, always a large-scale linear programming, the column generation algorithm is a commonly used and crucial method to solve it. In each iteration, this algorithm involves solving a linear programming subproblem and determining whether to terminate or generate a new column for inclusion in the subproblem. This iterative procedure of solving and checking continues throughout the algorithm. Nevertheless, since the checking subproblem is NP-hard, its solution significantly impacts the overall efficiency of the algorithm. In this paper, we innovatively convert the checking subproblem into a single-constrained shortest path (SCSP) subproblem. By exactly solving the SCSP subproblem, we can obtain the optimal solution to the checking subproblem with same or less computing time. Experimental results demonstrate that our algorithm achieves computational efficiency comparable to heuristic algorithms while outperforming other state-of-the-art algorithms by at least an order of magnitude.

An exact column generation algorithm for load balancing in capacity sharing networks

TL;DR

This work addresses load balancing in capacity sharing networks by formulating it as a maximum multi-commodity flow problem with per-technology capacity constraints. It introduces CGLAD, an exact column generation algorithm that converts the NP-hard optimality check into a single-constrained shortest path problem and solves it exactly using BiLAD and ExactBiLAD, guided by Lagrange duality. The method achieves optimal network throughput while complying with delay and technology-capacity constraints, and its performance is competitive with heuristic approaches while outperforming other exact methods on large instances. Numerical experiments on 200 randomly generated networks show CGLAD delivers strong throughput gains and strict delay adherence with computation times on par with state-of-the-art heuristics. Overall, CGLAD provides a robust benchmark and practical exactness for throughput optimization in capacity sharing networks.

Abstract

Capacity sharing networks are typical heterogeneous communication networks widely applied in information and communications technology (ICT) field. In such networks, resources like bandwidth, spectrum, computation and storage are shared among various communication services. Meanwhile, the issue of network congestion is always a prominent challenge. To handle network congestion essentially needs to solve the load balancing of networks. In this paper, for capacity sharing networks, we formulate their load balancing problem as a maximum multi-commodity flow problem. For such a problem, always a large-scale linear programming, the column generation algorithm is a commonly used and crucial method to solve it. In each iteration, this algorithm involves solving a linear programming subproblem and determining whether to terminate or generate a new column for inclusion in the subproblem. This iterative procedure of solving and checking continues throughout the algorithm. Nevertheless, since the checking subproblem is NP-hard, its solution significantly impacts the overall efficiency of the algorithm. In this paper, we innovatively convert the checking subproblem into a single-constrained shortest path (SCSP) subproblem. By exactly solving the SCSP subproblem, we can obtain the optimal solution to the checking subproblem with same or less computing time. Experimental results demonstrate that our algorithm achieves computational efficiency comparable to heuristic algorithms while outperforming other state-of-the-art algorithms by at least an order of magnitude.

Paper Structure

This paper contains 17 sections, 2 theorems, 15 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. System model and problem formulation
  3. System model
  4. Problem formulation
  5. Column generation algorithm
  6. Restricted master problem
  7. Optimality condition
  8. Generating new columns
  9. Explanation of new columns
  10. Complexity and convergence analysis
  11. Numerical experiments
  12. Generating topology of capacity sharing networks
  13. Numerical results
  14. The comparison of the computing time
  15. The comparison of network throughput
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that $\boldsymbol{\bar{f}}$ is the current solution obtained from RMP (2), the coefficient matrix of MCF (1) associated with $\boldsymbol{\bar{f}}$ is represented as $A_1=(B|N)$, and the coefficient vector in the objective function (1a) is represented as $\boldsymbol{c_1^{\top}}= (\boldsymbol holds.

Figures (5)

  • Figure 1: Topology diagram of capacity sharing networks with links of the identical color using the identical transmission technology.
  • Figure 2: The proposed column generation algorithm's procedure for the load balancing problem in capacity sharing networks.
  • Figure 3: The computing time of CGLAD compared with CGBB, CGDB and ViLBaS over 200 test problems.
  • Figure 4: The network throughput of CGLAD compared with CGBB, CGDB and ViLBaS over 200 test problems.
  • Figure 5: Path delay comparison between CGLAD and ViLBaS across 480 paths obtained from the first 30 test problems, with each data point representing the delay of an individual path and the delay threshold $\Delta_c = 30$ ms.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2