A Comprehensive Survey of Data Reduction Rules for the Maximum Weighted Independent Set Problem

TL;DR

This survey consolidates the landscape of exact data reduction rules for the Maximum Weight Independent Set problem and related problems, detailing a vast catalog of rules categorized by structural properties (low degree, neighborhood, clique, domination, struction, global, and further rules) and unified by a standard reduction framework (reduction name, reduced graph, offset, and reconstruction). It emphasizes practical applicability through a reference implementation and discusses the interplay between reductions and both exact branch-and-reduce solvers and heuristics, including reduce-and-peel strategies and recent meta-heuristics. The work highlights how reductions shrink instances, how some rules transform graphs rather than shrink them, and how global rules and CWIS relate to LP relaxations, with empirical observations on which reductions are commonly used in state-of-the-art solvers. By collecting and organizing these techniques, the paper aims to serve as a centralized, evolving resource for researchers and practitioners solving MWIS, MWVC, and MWC in practice. The contributions have immediate practical impact by guiding the design of more effective solvers and by providing a foundation for future reduction techniques and their integration into branch-and-reduce frameworks.

Abstract

The Maximum Weight Independent Set (MWIS) problem, as well as its related problems such as Minimum Weight Vertex Cover, are fundamental NP-hard problems with numerous practical applications. Due to their computational complexity, a variety of data reduction rules have been proposed in recent years to simplify instances of these problems, enabling exact solvers and heuristics to handle them more effectively. Data reduction rules are polynomial time procedures that can reduce an instance while ensuring that an optimal solution on the reduced instance can be easily extended to an optimal solution for the original instance. Data reduction rules have proven to be especially useful in branch-and-reduce methods, where successful reductions often lead to problem instances that can be solved exactly. This survey provides a comprehensive overview of data reduction rules for the MWIS problem. We also provide a reference implementation for these reductions. This survey will be updated as new reduction techniques are developed, serving as a centralized resource for researchers and practitioners.

Figures (7)

• Figure 1: This figure illustrates the history of MWC, MWVC, and MWIS solvers. The left axis gives a rough overview of publication years. A directed edge from a solver indicates a comparison made to another solver in the experimental evaluation. For example, the edge from MWCRedu to TSM-MWC indicates that MWCRedu used TSM-MWC in the experimental evaluation. The solvers that are highlighted in yellow are using data reductions.
• Figure 2: Different cases of Reduction \ref{['red:triangle']} with $\omega_x\leq\omega_y$. The status of a vertex after reducing is shown by its color, where green means included, red is excluded, and gray is folded.
• Figure 3: Different folding cases of Reduction \ref{['red:vShape']} with weights $\omega_x\leq\omega_y$.
• Figure 4: Illustration for Basic Single Edge; see Reduction \ref{['red:basicSE']}.
• Figure 5: Illustration for Extended Single Edge; see Reduction \ref{['red:extendedSE']}.

Theorems & Definitions (2)

• Remark 1
• Remark 2