Enabling Population-Based Architectures for Neural Combinatorial Optimization

TL;DR

This work addresses the gap between neural combinatorial optimization (NCO) and population-based search by introducing PB-NCO, a framework that operates on sets of solutions. It proposes two key components: contextual neural improvement (cNI), which uses a shared memory to coordinate local improvements, and conditioned neural constructive (cNC), which generates new candidates with explicit quality-diversity control via an exploration weight. The approach achieves competitive or superior results on Maximum Cut and Maximum Independent Set benchmarks, demonstrates enhanced diversity, and provides thorough ablations and analyses of design choices. By showing how neural policies can coordinate population dynamics, PB-NCO opens a bridge between learned optimization and classical population-based search, with practical implications for scalable, high-throughput combinatorial optimization.

Abstract

Neural Combinatorial Optimization (NCO) has mostly focused on learning policies, typically neural networks, that operate on a single candidate solution at a time, either by constructing one from scratch or iteratively improving it. In contrast, decades of work in metaheuristics have shown that maintaining and evolving populations of solutions improves robustness and exploration, and often leads to stronger performance. To close this gap, we study how to make NCO explicitly population-based by learning policies that act on sets of candidate solutions. We first propose a simple taxonomy of population awareness levels and use it to highlight two key design challenges: (i) how to represent a whole population inside a neural network, and (ii) how to learn population dynamics that balance intensification (generating good solutions) and diversification (maintaining variety). We make these ideas concrete with two complementary tools: one that improves existing solutions using information shared across the whole population, and the other generates new candidate solutions that explicitly balance being high-quality with diversity. Experimental results on Maximum Cut and Maximum Independent Set indicate that incorporating population structure is advantageous for learned optimization methods and opens new connections between NCO and classical population-based search.

Figures (10)

• Figure 1: High-level encoder–decoder architecture used in Neural Combinatorial Optimization. First (1) graph features (node/edge) are extracted, the encoder then produces node embeddings $h$ and edge embeddings $e$, projecting features to a $d$-dimensional space (2), and these are processed by a GNN through $L$ layers (3). The last layer embeddings are then used by a decoder (4) to produce action logits, that form action probabilities (5) that modify the current solution (6).
• Figure 2: Three levels of population awareness in neural population-based methods.(a) Independent: Each solution $s_t^{(i)}$ is improved in isolation. The policy is applied in parallel to multiple starting points, but there is no exchange of information between runs (equivalent to multi-start NCO). (b) Contextual: Each solution is updated based on its own state and an external context signal. This context can be global (e.g., a shared memory) or directed (e.g., peer-to-peer communication from specific neighbors). (c) Joint: The model reasons jointly about multiple solutions at once, allowing explicit pairwise or group interactions across the population.
• Figure 3: Overview of the proposed population-based NCO frameworks.(a) A contextual neural improvement policy (cNI) applies learned local moves to each solution, using information retrieved from a shared population memory of visited candidates. (b) A conditioned neural constructive policy (cNC) samples new candidates given a conditioning set of solutions $\mathcal{K}$ and an exploration weight $\omega$, which controls the trade-off between solution quality and diversity. (c) PB-NCO combines both components in a population loop, alternating cNI-driven improvement of existing solutions with cNC-driven restarts.
• Figure 4: Anytime performance against baselines. Solid lines denote the average best-so-far ratio (current objective / reference objective); shaded bands denote the standard deviation over five runs. Results are shown for both MC and MIS in ER700-800 and RB800-1200 graphs.
• Figure 5: Anytime performance against metaheuristics. Solid lines denote the average best-so-far ratio (current objective / reference objective); shaded bands denote the standard deviation over five runs. Results are shown for both MC and MIS in ER700-800 graphs.