Distribution-Aware Algorithm Design with LLM Agents

TL;DR

This paper introduces a distribution-aware framework for designing executable solver code using LLMs, optimizing both correctness and runtime based on learned hints from samples.

Contribution

It formalizes the concept of solver hints, proves generalization guarantees, and empirically demonstrates significant improvements in speed and quality across multiple problem distributions.

Findings

Synthesized solvers achieve high solution quality (mean normalized 0.971).

Significant speedups over heuristics, Gurobi, and exact solvers (up to 342x faster).

Effective hints lead to distribution-specific computational scale reduction.

Abstract

We study learning when the learned object is executable solver code rather than a predictor. In this setting, correctness is not enough: two solvers may both return valid solutions on the deployment distribution while differing substantially in runtime. Given samples from an unknown task distribution, the learner returns code evaluated on fresh instances by both solution quality and execution time. Our central abstraction is a \emph{solver hint}: reusable structure inferred from samples and compiled into specialized solver code. We prove that the empirically fastest sample-consistent solver from a fixed library generalizes in both correctness and runtime, and that statistically identifiable hints can be recovered and compiled from polynomially many samples. Empirically, we instantiate the framework with LLM code agents on \(21\) structured combinatorial-optimization target distributions across seven problem classes. The synthesized solvers reach mean normalized quality \(0.971\), improve by \(+0.224\) over the average heuristic pool and by \(+0.098\) over the highest-quality heuristic, and are \(336.9\times\), \(342.8\times\), and \(16.1\times\) faster than the quality-best heuristic, Gurobi, and the selected time-limited exact backend, respectively. On released PACE 2025 Dominating Set private instances, the synthesized solver is valid on all \(100\) graphs and runs about two orders of magnitude faster than top competition solvers, with a moderate quality gap. Inspection shows that many gains come from changing the computational scale: replacing ambient exponential search or general-purpose optimization with compiled distribution-specific computation.