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Accelerating Reductions Using Graph Neural Networks and a New Concurrent Local Search for the Maximum Weight Independent Set Problem

Ernestine Großmann, Kenneth Langedal, Christian Schulz

TL;DR

This work tackles the NP-hard Maximum Weight Independent Set problem by combining powerful data-reduction preprocessing with graph neural network screening and a novel concurrent search strategy. LearnAndReduce uses seven new reductions plus GNN-based early screening to apply expensive rules efficiently, supported by a labeled training dataset and three GNN architectures, notably LearnAndReduce (LR). The CHILS algorithm, built on a Baseline local search and a concurrent Difference-Core (D-Core) approach, achieves state-of-the-art results across real-world and vehicle-routing benchmarks, with substantial parallel speedups up to 104x on 128 cores. Overall, the combination of exact preprocessing and parallelizable heuristic search significantly improves MWIS solving capability and offers directions for applying similar strategies to related combinatorial problems.

Abstract

The Maximum Weight Independent Set problem is a fundamental NP-hard problem in combinatorial optimization with several real-world applications. Given an undirected vertex-weighted graph, the problem is to find a subset of the vertices with the highest possible weight under the constraint that no two vertices in the set can share an edge. An important part of solving this problem in both theory and practice is data reduction rules, which several state-of-the-art algorithms rely on. However, the most complicated rules are often not used in applications since the time needed to check them exhaustively becomes infeasible. In this work, we introduce three main results. First, we introduce several new data reduction rules and evaluate their effectiveness on real-world data. Second, we use a machine learning screening algorithm to speed up the reduction phase, thereby enabling more complicated rules to be applied. Our screening algorithm consults a Graph Neural Network oracle to decide if the probability of successfully reducing the graph is sufficiently large. For this task, we provide a dataset of labeled vertices for use in supervised learning. We also present the first results for this dataset using established Graph Neural Network architectures. Third, we present a new concurrent metaheuristic called Concurrent Difference-Core Heuristic. On the reduced instances, we use our new metaheuristic combined with iterated local search, called CHILS (Concurrent Hybrid Iterated Local Search). For this iterated local search, we provide a new implementation specifically designed to handle large graphs of varying densities. CHILS outperforms the current state-of-the-art on all commonly used benchmark instances, especially the largest ones.

Accelerating Reductions Using Graph Neural Networks and a New Concurrent Local Search for the Maximum Weight Independent Set Problem

TL;DR

This work tackles the NP-hard Maximum Weight Independent Set problem by combining powerful data-reduction preprocessing with graph neural network screening and a novel concurrent search strategy. LearnAndReduce uses seven new reductions plus GNN-based early screening to apply expensive rules efficiently, supported by a labeled training dataset and three GNN architectures, notably LearnAndReduce (LR). The CHILS algorithm, built on a Baseline local search and a concurrent Difference-Core (D-Core) approach, achieves state-of-the-art results across real-world and vehicle-routing benchmarks, with substantial parallel speedups up to 104x on 128 cores. Overall, the combination of exact preprocessing and parallelizable heuristic search significantly improves MWIS solving capability and offers directions for applying similar strategies to related combinatorial problems.

Abstract

The Maximum Weight Independent Set problem is a fundamental NP-hard problem in combinatorial optimization with several real-world applications. Given an undirected vertex-weighted graph, the problem is to find a subset of the vertices with the highest possible weight under the constraint that no two vertices in the set can share an edge. An important part of solving this problem in both theory and practice is data reduction rules, which several state-of-the-art algorithms rely on. However, the most complicated rules are often not used in applications since the time needed to check them exhaustively becomes infeasible. In this work, we introduce three main results. First, we introduce several new data reduction rules and evaluate their effectiveness on real-world data. Second, we use a machine learning screening algorithm to speed up the reduction phase, thereby enabling more complicated rules to be applied. Our screening algorithm consults a Graph Neural Network oracle to decide if the probability of successfully reducing the graph is sufficiently large. For this task, we provide a dataset of labeled vertices for use in supervised learning. We also present the first results for this dataset using established Graph Neural Network architectures. Third, we present a new concurrent metaheuristic called Concurrent Difference-Core Heuristic. On the reduced instances, we use our new metaheuristic combined with iterated local search, called CHILS (Concurrent Hybrid Iterated Local Search). For this iterated local search, we provide a new implementation specifically designed to handle large graphs of varying densities. CHILS outperforms the current state-of-the-art on all commonly used benchmark instances, especially the largest ones.

Paper Structure

This paper contains 24 sections, 2 theorems, 1 equation, 9 figures, 8 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related Work
  4. Exact Methods
  5. Heuristic Methods
  6. Learn and Reduce - GNN Guided Preprocessing
  7. New Data Reduction Rules
  8. GNN Models
  9. Training Data Generation
  10. The Learn and Reduce Approach
  11. The CHILS Algorithm
  12. High-Level Description
  13. Baseline Local Search
  14. CHILS
  15. Experimental Evaluation
  16. ...and 9 more sections

Key Result

Lemma 4

Let $v \in V$ and $C$ be the set of independent sets in $G[N(v)]$ with higher weight than $v$. Then, we can construct an instance $G'$ by removing $N[v]$ and for each solution $c \in C$, add a vertex $v_c$ with weight $\omega(c)-\omega(v)$. All these solution vertices are connected to form a clique.

Figures (9)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4: Training loss for each architecture using the unconfined, critical set, and generalized fold reduction rules on the test set. Note that the scaling of the y-axes differ.
  • Figure 5: Accuracy, coverage, and the fraction remaining after the screening for each architecture using the heavy set reduction rule. The values shown here are computed based on the test dataset. Note that the scaling of the y-axes differ.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Remark 6