Gradient Descent Learning Based on Reinforcement Learning Strategy for Solving Graph Coloring Problem

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Introduction

The Graph Coloring Problem (GCP) is a fundamental combinatorial optimization challenge that involves assigning colors to the vertices of a graph such that no two adjacent vertices share the same color while minimizing the total number of colors used. As an NP-hard problem, GCP finds applications in various domains, including wireless network frequency allocation, task scheduling, map coloring, and constraint satisfaction problems. Traditional methods for solving GCP, such as greedy algorithms, heuristic search techniques, and evolutionary algorithms, often struggle with high computational complexity and a tendency to converge to suboptimal solutions, particularly in large-scale or structurally complex graphs.

To address these limitations, this paper introduces a novel approach that leverages reinforcement learning strategies (RLS) combined with gradient descent optimization to solve GCP. By reformulating GCP as a policy optimization problem within a reinforcement learning (RL) framework, the proposed method dynamically adjusts color assignments to minimize conflicts and reduce the number of colors used. The approach demonstrates superior performance compared to conventional heuristic algorithms, particularly in high-dimensional and complex constraint scenarios.

Problem Formulation and Reinforcement Learning Framework

Graph Coloring Problem as a Combinatorial Optimization Challenge

GCP is formally defined on an undirected graph ( G = (V, E) ), where ( V ) represents the set of vertices and ( E ) denotes the edges connecting these vertices. The objective is to assign a color ( c_i ) from a set ( C ) to each vertex ( v_i \in V ) such that for every edge ( (v_i, v_j) \in E ), ( c_i \neq c_j ). The optimization goal is to minimize ( |C| ), the total number of distinct colors used.

Traditional approaches often rely on heuristic rules or stochastic search strategies, which may perform well in specific cases but fail to generalize across diverse graph structures. These methods are particularly limited in their ability to explore the solution space globally, often getting trapped in local optima.

Reinforcement Learning Framework for GCP

The proposed method transforms GCP into an RL problem by defining four key components: state, action, policy, and reward.

State Representation

The state ( s_t ) at time step ( t ) captures the current coloring configuration of the graph. Two primary state representation methods are employed:

  1. Vector Representation: A vector of length ( |V| ) where each element ( s_t(i) ) indicates the color assigned to vertex ( v_i ). Uncolored vertices are marked with a special value (e.g., 0). This method is simple and intuitive but may struggle with large-scale graphs.
  2. Matrix Representation: A binary matrix of size ( |V| \times |C| ), where ( s_t(i, j) = 1 ) if vertex ( v_i ) is assigned color ( j ). This approach provides a more detailed view of the graph’s coloring state, making it suitable for complex or dense graphs.

Action Space

An action ( a_t ) consists of selecting a vertex and assigning it a color. Specifically, ( a_t ) can be represented as a tuple ( (v_i, c_j) ), where ( v_i ) is the vertex to be colored, and ( c_j ) is the chosen color. Alternatively, actions can be defined as updates to existing color assignments, modifying the color of a vertex from its current value to a new one.

Policy Network

The policy ( \pi(a_t | s_t; \theta) ) is a probability distribution over possible actions given the current state, parameterized by ( \theta ). A Graph Neural Network (GNN) is used to implement the policy network, leveraging its ability to capture structural information from the graph. The GNN aggregates features from neighboring vertices, enabling the policy to make informed decisions based on local and global graph properties.

Reward Function

The reward ( R_t ) is designed to guide the learning process by quantifying the quality of each action. The primary reward is derived from the reduction in color conflicts:

[ R_t = -\Delta L(C) = -(L(C_{t+1}) – L(C_t)) ]

where ( L(C) ) counts the number of conflicting edges (i.e., edges with identically colored vertices). A negative reward is given if conflicts increase, while a positive reward is provided if conflicts decrease.

To further optimize color usage, a composite reward function is introduced:

[ R_t = -\alpha \Delta L(C) – \beta |C| ]

Here, ( \alpha ) and ( \beta ) are weighting parameters that balance the trade-off between minimizing conflicts and reducing the number of colors used.

Policy Gradient Method for Solving GCP

Overview of Policy Gradient Optimization

Policy gradient methods directly optimize the policy ( \pi(a_t | s_t; \theta) ) by adjusting its parameters ( \theta ) to maximize the expected cumulative reward ( J(\theta) ). The gradient of ( J(\theta) ) is estimated using sampled trajectories, allowing the policy to improve iteratively.

The REINFORCE algorithm, a Monte Carlo policy gradient method, is employed here. It updates the policy parameters using:

[ \theta \leftarrow \theta + \alpha \sum_{t=0}^T \nabla \log \pi(a_t | s_t; \theta) \hat{R}_t ]

where ( \hat{R}_t ) is the cumulative discounted reward from time step ( t ) onward, and ( \alpha ) is the learning rate.

Variance Reduction and Baseline Subtraction

To stabilize training, a baseline ( b(s_t) ) is subtracted from the reward to reduce variance in gradient estimates. The baseline, often chosen as the state value function ( V(s_t) ), does not bias the gradient but significantly improves convergence.

Training Process

  1. Initialization: The policy network parameters ( \theta ) are randomly initialized, and the initial graph state ( s_0 ) is set (e.g., all vertices uncolored).
  2. Trajectory Sampling: The current policy is used to generate sequences of states, actions, and rewards.
  3. Gradient Computation: For each trajectory, gradients are computed with respect to the policy parameters.
  4. Parameter Update: The policy is updated using gradient ascent to maximize expected rewards.
  5. Termination: The process repeats until convergence or a maximum number of iterations is reached.

Theoretical Analysis of Policy Optimality

The proposed method ensures convergence to a near-optimal policy under the following assumptions:

  1. Sufficient Exploration: The policy must explore the state-action space adequately to avoid local optima.
  2. Appropriate Learning Rate: A well-tuned learning rate ensures stable convergence without premature stopping.

The Bellman optimality equation underpins the theoretical foundation, guaranteeing that the policy improves monotonically toward maximizing cumulative rewards. Compared to traditional heuristic methods, which rely on local search and may stagnate, reinforcement learning explores the solution space more effectively, balancing exploration and exploitation.

Experimental Results

Experimental Setup

The proposed method is evaluated on benchmark datasets from the DIMACS graph coloring challenge, including DSJC and le450 series. Performance is compared against state-of-the-art algorithms:

  • Tabucol+: An enhanced tabu search algorithm.
  • PLHEAD: A hybrid evolutionary algorithm with reinforcement learning.
  • PLSCOL: A probability-based local search method.
  • MEA-GCP: A membrane evolutionary algorithm.
  • TensCol: A tensor-based optimization framework.

Performance Evaluation

The experiments measure two key metrics:

  1. Solution Quality: The minimum number of colors ( c ) required to achieve a valid coloring.
  2. Success Rate: The percentage of runs (out of 20 trials) that achieve a valid coloring with ( c ) colors.

Key Findings

  1. Optimal Solutions: The proposed RLS method matches or outperforms existing algorithms in 23 out of 24 instances. For example:
    • On DSJC500.1, RLS achieves a valid 12-coloring with 100% success, comparable to PLHEAD and MEA-GCP. • On DSJC1000.5, RLS finds an 83-coloring with 100% success, while PLHEAD achieves 90% success.
  2. Success Rates: RLS consistently demonstrates higher success rates, particularly in complex graphs. For instance:
    • On R250.5, RLS attains a 65-coloring with 100% success, whereas Tabucol+ only reaches 10%. • On flat1000_50_0, RLS solves the problem with 50 colors and 100% success, outperforming other methods.
  3. Adaptability: RLS excels in both sparse and dense graphs, showcasing its robustness across varying graph structures.

Comparative Analysis

  • Tabucol+: Struggles with global exploration due to its reliance on local search.
  • PLHEAD: Performs well but requires extensive tuning of heuristic rules.
  • PLSCOL: Computationally expensive due to probabilistic matrix updates.
  • MEA-GCP: High complexity from managing multiple membrane structures.
  • TensCol: Incurs unnecessary overhead by handling both GCP and equitable coloring.

In contrast, RLS directly optimizes the policy through gradient descent, enabling efficient exploration and exploitation without heuristic dependencies.

Conclusion

This paper presents a reinforcement learning-based approach to solving the Graph Coloring Problem, leveraging policy gradient methods to dynamically optimize color assignments. By framing GCP as an RL task, the method effectively balances conflict reduction and color minimization, demonstrating superior performance across diverse graph instances.

Key advantages include:

  • Global Exploration: Avoids local optima through strategic state-action exploration.
  • Adaptability: Performs well on both small and large-scale graphs.
  • Scalability: Handles complex constraints efficiently.

Future work will focus on:

  1. Enhancing the policy network with advanced graph features for better structural awareness.
  2. Incorporating meta-learning to improve generalization across graph types.
  3. Applying transfer learning to reduce training time on new instances.

This research establishes a promising direction for solving combinatorial optimization problems using reinforcement learning, offering a flexible and powerful alternative to traditional heuristic methods.

DOI: 10.19734/j.issn.1001-3695.2024.09.0330

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