[Paper Review] ILP formulations for the two-stage stochastic Steiner tree problem

Bernd Zey|arXiv (Cornell University)|Nov 14, 2016

VLSI and FPGA Design Techniques29 references3 citations

TL;DR

This paper presents and compares multiple integer linear programming (ILP) formulations for the two-stage stochastic Steiner tree problem (SSTP), focusing on cut-based, flow-based, and directed formulations. It establishes theoretical equivalence between key formulations and demonstrates that directed flow formulations yield stronger LP relaxations, significantly improving solution quality for stochastic network design under uncertainty.

ABSTRACT

We give an overview of new and existing cut- and flow-based ILP formulations for the two-stage stochastic Steiner tree problem and compare the strength of the LP relaxations.

Motivation & Objective

• To develop and compare new and existing ILP formulations for the two-stage stochastic Steiner tree problem (SSTP), including cut-based, flow-based, and directed formulations.
• To analyze the strength of LP relaxations across different formulations to identify which yield tighter bounds and better performance in practice.
• To establish theoretical equivalence between key formulations, particularly between directed flow and directed cut formulations, ensuring solution consistency.
• To investigate the impact of structural constraints—such as requiring a rooted first-stage tree—on formulation design and solution quality.
• To provide a comprehensive framework for solving stochastic network design problems under uncertainty using strong, scalable ILP models.

Proposed method

• Proposes a new directed cut formulation (rSSTP_dc2) that models connectivity via directed cuts in the first stage and in each scenario.
• Introduces a directed flow formulation (rSSTP_df) that uses flow variables to model connectivity from a root node to all terminals in each scenario.
• Establishes equivalence between the directed flow formulation and the second directed cut formulation through projection, proving that both yield the same LP relaxation strength.
• Uses max-flow min-cut duality to validate feasibility of flow-based solutions and ensure that cut constraints are satisfied.
• Employs variable transformations and projection techniques to compare formulations and verify that constraints such as capacity and connectivity are preserved.
• Analyzes the impact of edge cost assumptions (e.g., c_e^0 < c_e^*) in the context of rooted vs. non-rooted SSTP, showing that such assumptions may not hold in the rooted variant.

Experimental results

Research questions

• RQ1 How do different ILP formulations for the two-stage stochastic Steiner tree problem compare in terms of LP relaxation strength?
• RQ2 What is the theoretical relationship between directed flow and directed cut formulations in the context of the rooted stochastic Steiner tree problem?
• RQ3 Can directed formulations provide stronger LP bounds than undirected or semi-directed formulations in stochastic network design?
• RQ4 Why do standard assumptions like c_e^0 < c_e^* fail in the rooted stochastic Steiner tree problem?
• RQ5 To what extent do structural constraints (e.g., requiring a first-stage tree) affect the design and strength of ILP formulations?

Key findings

• The directed flow formulation (rSSTP_df) and the second directed cut formulation (rSSTP_dc2) are mathematically equivalent in terms of their projected solution space, as proven by mutual inclusion of their projections.
• The directed flow formulation yields strictly stronger LP relaxations than the undirected cut and flow formulations, indicating better bound quality for branch-and-bound solvers.
• The directed cut formulation (rSSTP_dc2) is shown to be equivalent to the directed flow formulation through a constructive proof involving flow routing and max-flow min-cut duality.
• The assumption c_e^0 < c_e^* does not hold in the rooted SSTP, as demonstrated by a counterexample where disabling an edge in the first stage leads to suboptimal solutions.
• The paper confirms that the rooted SSTP requires different modeling approaches than the non-rooted version, particularly in handling first-stage connectivity and cost assumptions.
• Theoretical results show that all constraints in the formulations—including capacity, connectivity, and variable bounds—are preserved under variable transformations and projections.