combinatorial-optimization

google Sep 29, 2025

AI as a research partner: Advancing theoretical computer science with AlphaEvolve (opens in new tab)

AlphaEvolve, an LLM-powered coding agent developed by Google DeepMind, facilitates mathematical discovery by evolving code to find complex combinatorial structures that are difficult to design manually. By utilizing a "lifting" technique, the system discovers finite structures that can be plugged into existing proof frameworks to establish new universal theorems in complexity theory. This methodology has successfully produced state-of-the-art results for the MAX-4-CUT problem and tightened bounds on the hardness of certifying properties in random graphs. ## The Role of AlphaEvolve in Mathematical Research * The system uses an iterative feedback loop to morph code snippets, evaluating the resulting mathematical structures and refining the code toward more optimal solutions. * AlphaEvolve operates as a tool-based assistant that generates specific proof elements, which can then be automatically verified by computer programs to ensure absolute mathematical correctness. * By focusing on verifiable finite structures, the agent overcomes the common "hallucination" issues of LLMs, as the final output is a computationally certified object rather than a speculative text-based proof. ## Bridging Finite Discovery and Universal Statements through Lifting * Theoretical computer science often requires proofs that hold true for all problem sizes ($\forall n$), a scale that AI systems typically struggle to address directly. * The "lifting" technique treats a proof as a modular structure where a specific finite component—such as a combinatorial gadget—can be replaced with a more efficient version while keeping the rest of the proof intact. * When AlphaEvolve finds a superior finite structure, the improvement is "lifted" through the existing mathematical framework to yield a stronger universal theorem without requiring a human to redesign the entire logical architecture. ## Optimizing Gadget Reductions and MAX-k-CUT * Researchers applied the agent to "gadget reductions," which are recipes used to map known intractable problems to new ones to prove computational hardness (NP-hardness). * AlphaEvolve discovered complex gadgets that were previously unknown because they were too intricate for researchers to construct by hand. * These discoveries led to a new state-of-the-art inapproximability result for the MAX-4-CUT problem, defining more precise limits on how accurately the problem can be solved by any efficient algorithm. ## Advancing Average-Case Hardness in Random Graphs * The agent was tasked with uncovering structures related to the average-case hardness of certifying properties within random graphs. * By evolving better combinatorial structures for these specific instances, the team was able to tighten existing mathematical bounds, providing a clearer picture of when certain graph properties become computationally intractable to verify. This research demonstrates that LLM-based agents can serve as genuine research partners by focusing on the discovery of verifiable, finite components within broader theoretical frameworks. For researchers in mathematics and computer science, this "lifting" approach provides a practical roadmap for using AI to solve bottleneck problems that were previously restricted by the limits of manual construction.

combinatorial-optimization ai llm gemini+5

google Jun 5, 2025

Optimizing LLM-based trip planning (opens in new tab)

Google Research has developed a hybrid planning system that combines Large Language Models (LLMs) with traditional optimization algorithms to solve complex trip-planning tasks. While LLMs excel at interpreting qualitative user preferences—such as a desire for "lesser-known museums"—they often struggle with hard quantitative constraints like travel logistics and fluctuating opening hours. By using an LLM to generate an initial draft and a secondary algorithm to refine it against real-world data, the system produces itineraries that are both highly personalized and logistically feasible. ## The Hybrid Planning Architecture * The process begins with a Gemini model generating an initial trip plan based on the user's natural language query, identifying specific activities and their perceived importance. * This draft is grounded using live data, incorporating up-to-date opening hours, transit schedules, and travel times between locations. * Search backends simultaneously retrieve alternative activities to serve as potential substitutes if the LLM's original suggestions prove logistically impossible. ## Two-Stage Optimization Algorithm * The first stage focuses on single-day scheduling, using dynamic programming and exhaustive search to find the most efficient sequence for subsets of activities. * Each potential daily schedule is assigned a quality score based on its feasibility and how closely it aligns with the LLM's original intent. * The second stage addresses the multi-day itinerary as a weighted variant of the "set packing problem," which ensures that activities do not overlap across different days. * Because multi-day optimization is NP-complete, the system employs local search heuristics to swap activities between days, iteratively improving the total score until the plan converges. ## Balancing Intent and Feasibility * In practical testing, the system demonstrated a superior ability to handle nuanced requests, such as finding "lesser-known" museums in NYC, which traditional retrieval systems often fail by suggesting famous landmarks like the Met. * The optimization layer specifically corrects geographical inefficiencies, such as the LLM suggesting a "zig-zag" route across San Francisco, by regrouping activities into logical clusters to minimize travel time. * The system maintains the "spirit" of the LLM's creative suggestions—like visiting a specific scenic viewpoint—while ensuring the user doesn't arrive after the gates have closed. This hybrid approach suggests that the most reliable AI planning tools do not rely on LLMs in isolation. By using LLMs as creative engines for intent interpretation and delegating logistical verification to rigid algorithmic frameworks, developers can create tools that are both imaginative and practically dependable.

combinatorial-optimization ai llm gemini+4