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Grouping Strategies on Two-Phase Methods for Bi-objective Combinatorial Optimization

Felipe O. Mota, Luís Paquete, Daniel Vanderpooten

TL;DR

This work tackles the computational bottleneck of two-phase bi-objective combinatorial optimization by reducing redundant second-phase ranking runs through grouping adjacent search zones and via a coverage-based implicit grouping. It also provides an optimal grouping method based on a shortest-path formulation over a grouping graph and develops multiple a priori and dynamic strategies that leverage measures like the ND bound and group angle, plus extended coverage to propagate information across groups. Empirical results on BOMST show substantial reductions in counted solutions and that dynamic strategies with extended coverage consistently outperform baseline ranking, with typical optimal groupings concentrating in group sizes of 2–3. The findings highlight practical impact for parallelizable MOCO workflows and generalize to other problems where a ranking-based second phase is used, offering guidance on when and how to group search zones to minimize redundant exploration.

Abstract

Two-phase methods are commonly used to solve bi-objective combinatorial optimization problems. In the first phase, all extreme supported nondominated points are generated through a dichotomic search. This phase also allows the identification of search zones that may contain other nondominated points. The second phase focuses on exploring these search zones to locate the remaining points, which typically accounts for most of the computational cost. Ranking algorithms are frequently employed to explore each zone individually, but this approach leads to redundancies, causing multiple visits to the same solutions. To mitigate these redundancies, we propose several strategies that group adjacent zones, allowing a single run of the ranking algorithm for the entire group. Additionally, we explore an implicit grouping approach based on a new concept of coverage. Our experiments on the Bi-Objective Spanning Tree Problem demonstrate the beneficial impact of these grouping strategies when combined with coverage.

Grouping Strategies on Two-Phase Methods for Bi-objective Combinatorial Optimization

TL;DR

This work tackles the computational bottleneck of two-phase bi-objective combinatorial optimization by reducing redundant second-phase ranking runs through grouping adjacent search zones and via a coverage-based implicit grouping. It also provides an optimal grouping method based on a shortest-path formulation over a grouping graph and develops multiple a priori and dynamic strategies that leverage measures like the ND bound and group angle, plus extended coverage to propagate information across groups. Empirical results on BOMST show substantial reductions in counted solutions and that dynamic strategies with extended coverage consistently outperform baseline ranking, with typical optimal groupings concentrating in group sizes of 2–3. The findings highlight practical impact for parallelizable MOCO workflows and generalize to other problems where a ranking-based second phase is used, offering guidance on when and how to group search zones to minimize redundant exploration.

Abstract

Two-phase methods are commonly used to solve bi-objective combinatorial optimization problems. In the first phase, all extreme supported nondominated points are generated through a dichotomic search. This phase also allows the identification of search zones that may contain other nondominated points. The second phase focuses on exploring these search zones to locate the remaining points, which typically accounts for most of the computational cost. Ranking algorithms are frequently employed to explore each zone individually, but this approach leads to redundancies, causing multiple visits to the same solutions. To mitigate these redundancies, we propose several strategies that group adjacent zones, allowing a single run of the ranking algorithm for the entire group. Additionally, we explore an implicit grouping approach based on a new concept of coverage. Our experiments on the Bi-Objective Spanning Tree Problem demonstrate the beneficial impact of these grouping strategies when combined with coverage.

Paper Structure

This paper contains 21 sections, 1 theorem, 12 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definitions
  3. Related work
  4. Groups and groupings
  5. Optimal grouping
  6. Grouping strategies
  7. Grouping measures
  8. Group angle
  9. Group ND bound
  10. A priori strategy
  11. Merge-based grouping
  12. Fixed-size variant
  13. Greedy variant
  14. Splitting-based grouping
  15. Dynamic strategy
  16. ...and 6 more sections

Key Result

Proposition 4.1

Let $y^i, \ldots, y^{j+1}$ be $j-i+2$ consecutive supported points that are collinear and that define $j-i+1$ adjacent triangles $\Delta^i,\ldots,\Delta^j$, $j-i>0$. Then the computational effort required for group ${\mathcal{G}}^{{i}{j}}$ coincides with the computational effort required for one of

Figures (5)

  • Figure 1: Illustration of a triangle $\Delta^{{i}}$ and its upper bound $u^{*}$ (left) and the updated upper bound $u^{*}$ given that point $y$ was found in $\Delta^{{i}}$ (right)
  • Figure 2: Illustration of group ${\mathcal{G}}^{{i}{j}}$, its upper bound ($u^{*}$) and starting solution ($y^*$).
  • Figure 3: Grouping triangles defined by supported points far from collinear may not be beneficial.
  • Figure 4: Example of the angle measure in groups of size 2 (left) and in the general case (right).
  • Figure 5: Example of covered, partially covered, and uncovered triangles

Theorems & Definitions (2)

  • Proposition 4.1
  • proof