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Robustifying networks for flow problems against edge failure

Artyom Klyuchikov, Roland Hildebrand, Sergei Protasov, Alexander Rogozin, Alexei Chernov

TL;DR

This paper addresses robustness of multi-commodity network flows under edge failures by formalizing worst-case degradation when up to $q$ edges fail and by proposing budgeted robustification to mitigate this risk. It develops two solution paradigms: (i) enlarging the LP to capture robust scenarios, and (ii) a three-level optimization with the lower level consisting of standard LPs solved by a dual simplex method with warm starts, enabling efficient handling of discrete failure configurations. The authors introduce a budget-based robustification framework that allocates additional capacity with total increase $B$ to minimize the worst-case impact of edge deletions, yielding a convex outer problem that can be tackled with gradient-based methods. Numerical experiments on SNDLib telecom networks demonstrate that the warm-start dual simplex approach is competitive and often superior to alternative solvers, suggesting practical applicability to network reliability and resilience planning.

Abstract

We consider the robust version of a multi-commodity network flow problem. The robustness is defined with respect to the deletion, or failure, of edges. While the flow problem itself is a polynomially-sized linear program, its robust version is a saddle-point problem with discrete variables. We present two approaches for the solution of the robust network flow problem. One way is to formulate the problem as a bigger linear program. The other is to solve a multi-level optimization problem, where the linear programs appearing at the lower level can be solved by the dual simplex method with a warm start. We then consider the problem of robustifying the network. This is accomplished by optimally using a fixed budget for strengthening certain edges, i.e., increasing their capacity. This problem is solved by a sequence of linear programs at the upper level, while at the lower levels the mentioned dual simplex algorithm is employed.

Robustifying networks for flow problems against edge failure

TL;DR

This paper addresses robustness of multi-commodity network flows under edge failures by formalizing worst-case degradation when up to edges fail and by proposing budgeted robustification to mitigate this risk. It develops two solution paradigms: (i) enlarging the LP to capture robust scenarios, and (ii) a three-level optimization with the lower level consisting of standard LPs solved by a dual simplex method with warm starts, enabling efficient handling of discrete failure configurations. The authors introduce a budget-based robustification framework that allocates additional capacity with total increase to minimize the worst-case impact of edge deletions, yielding a convex outer problem that can be tackled with gradient-based methods. Numerical experiments on SNDLib telecom networks demonstrate that the warm-start dual simplex approach is competitive and often superior to alternative solvers, suggesting practical applicability to network reliability and resilience planning.

Abstract

We consider the robust version of a multi-commodity network flow problem. The robustness is defined with respect to the deletion, or failure, of edges. While the flow problem itself is a polynomially-sized linear program, its robust version is a saddle-point problem with discrete variables. We present two approaches for the solution of the robust network flow problem. One way is to formulate the problem as a bigger linear program. The other is to solve a multi-level optimization problem, where the linear programs appearing at the lower level can be solved by the dual simplex method with a warm start. We then consider the problem of robustifying the network. This is accomplished by optimally using a fixed budget for strengthening certain edges, i.e., increasing their capacity. This problem is solved by a sequence of linear programs at the upper level, while at the lower levels the mentioned dual simplex algorithm is employed.

Paper Structure

This paper contains 20 sections, 1 theorem, 26 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nominal problem formulation
  3. Network model
  4. Throughput
  5. Relation to load balance
  6. Average latency
  7. Dual simplex method with warm start
  8. Computation of derivatives of the objective function
  9. Adding constraints
  10. Network robustification problem
  11. Robust throughput
  12. Robust latency
  13. Throughput robustification
  14. Latency robustification
  15. Solution of the problems
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let all bandwidths be positive, i.e. $b > 0$ and let at least one of the demands be nonzero. Then problems eq:throughput and eq:load_balance are equivalent in the sense that given a solution of eq:throughput we can immediately obtain a solution of eq:load_balance and vice versa.

Figures (2)

  • Figure 1: Optimization of robust latency
  • Figure 2: Optimization of robust throughput

Theorems & Definitions (3)

  • Lemma 2.1
  • proof
  • Remark 2.2