Hard Constraints Meet Soft Generation: Guaranteed Feasibility for LLM-based Combinatorial Optimization

TL;DR

This paper presents FALCON, a framework that guarantees 100% feasibility in LLM-based combinatorial optimization by combining grammar constraints, a repair layer, and adaptive sampling, along with a novel training method called BOPO.

Contribution

Introduces FALCON, a novel framework ensuring guaranteed feasibility in LLM-based CO solutions, and proposes BOPO for effective LLM training without human labels.

Findings

Achieves perfect feasibility across seven NP-hard problems.

Matches or exceeds state-of-the-art solution quality.

Provides theoretical guarantees and bounds for the proposed methods.

Abstract

Large language models (LLMs) have emerged as promising general-purpose solvers for combinatorial optimization (CO), yet they fundamentally lack mechanisms to guarantee solution feasibility which is critical for real-world deployment. In this work, we introduce FALCON, a framework that ensures 100\% feasibility through three key innovations: (i) \emph{grammar-constrained decoding} enforces syntactic validity, (ii) a \emph{feasibility repair layer} corrects semantic constraint violations, and (iii) \emph{adaptive Best-of-$N$ sampling} allocates inference compute efficiently. To train the underlying LLM, we introduce the Best-anchored Objective-guided Preference Optimization (BOPO) in LLM training, which weights preference pairs by their objective gap, providing dense supervision without human labels. Theoretically, we prove convergence for BOPO and provide bounds on repair-induced quality loss. Empirically, across seven NP-hard CO problems, FALCON achieves perfect feasibility while matching or exceeding the solution quality of state-of-the-art neural and LLM-based solvers.