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Iterative Belief Propagation for Sparse Combinatorial Optimization

Sam Reifenstein, Timothée Leleu

Abstract

In this note we study an iterative belief propagation (IBP) algorithm and demonstrate it's ability to solve sparse combinatorial optimization problems. Similar to simulated annealing (SA), our IBP algorithm attempts to sample from the Boltzmann distribution of the objective function but also uses belief propagation (BP) to improve convergence.

Iterative Belief Propagation for Sparse Combinatorial Optimization

Abstract

In this note we study an iterative belief propagation (IBP) algorithm and demonstrate it's ability to solve sparse combinatorial optimization problems. Similar to simulated annealing (SA), our IBP algorithm attempts to sample from the Boltzmann distribution of the objective function but also uses belief propagation (BP) to improve convergence.

Paper Structure

This paper contains 10 sections, 10 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction and Algorithm Overview
  2. Numerical Results
  3. Algorithm Details
  4. Algorithm Pseudo-code
  5. Selecting Sub-trees
  6. Belief Propagation
  7. Logarithmic form of BP
  8. Sampling from the Boltzmann Distribution with BP
  9. Computational Complexity
  10. Conclusion and Outlook

Figures (1)

  • Figure 1: Median, 1st percentile and best objective values found by SA and IBP with respect to total spin updates on Left: a Max-Cut instance, Middle: a Maximum Independent Set instance and Right: a sparse random QUBO instance. All three instance are generated from Erdős–Rényi random grpahs of size $N=2000$ and density $1\%$. The number of spin updates for IBP is defined as the total size of all sub-trees considered (average size pf sube-tree multiplied by number of iterations) while the number of spin update for SA is number of sweeps muptiplied by $N$. The same geometric annealing sechdule is used for $\beta$ for both algorithms.