Approximation Algorithms

googleFeb 10, 20263 min readCurated summary

Scheduling in a changing world: Maximizing throughput with time-varying capacity

The post presents scheduling algorithms for non-preemptive jobs when cloud capacity changes over time because of failures, maintenance, power limits, or higher-priority workloads. The goal is to maximize completed job value while respecting release times, deadlines, processing durations, and fluctuating parallel capacity. The research establishes the first constant-factor guarantees for several offline and online variants, including a 1/11 competitive ratio for a demanding common-deadline model. ## Scheduling with Time-Varying Capacity - A capacity profile specifies how many jobs can run simultaneously at each point in time. - Each job has: - A release time - A hard deadline - A processing duration - A weight or profit - Jobs must run continuously once started in the non-preemptive setting. - If capacity drops during execution, an interrupted job loses its progress. - The objective is to select and schedule jobs maximizing total completed weight. - The study considers: - **Offline scheduling**, where future jobs and capacity changes are known. - **Online scheduling**, where jobs arrive dynamically and decisions cannot be reversed. ## Offline Scheduling Results - The optimal problem is NP-hard, so the work focuses on approximation guarantees. - For unit-profit jobs, an earliest-finish-time Greedy algorithm achieves a **1/2-approximation**. - It completes at least half as many jobs as an optimal schedule. - This matches the classic guarantee for single-capacity scheduling. - For jobs with different weights, a primal-dual algorithm achieves a **1/4-approximation**. ## Why Online Non-Preemptive Scheduling Is Difficult - Online schedulers must commit without knowing future jobs. - Starting a long job can block many shorter jobs that arrive later. - Because each completed job may have equal value regardless of duration, one poor decision can sharply reduce throughput. - Consequently, standard non-preemptive online algorithms have competitive ratios approaching zero. ## Interruption with Restarts - An active job may be interrupted, but its completed work is discarded and the job can be retried later. - A modified earliest-finish-time Greedy algorithm achieves a **1/2 competitive ratio**. - This means it can guarantee at least half the throughput of an optimal schedule with complete knowledge of future arrivals. ## Interruption Without Restarts - If an interrupted job is permanently discarded, online scheduling becomes substantially harder. - In general, every online algorithm can be forced into decisions that prevent it from completing much future work. - The competitive ratio again approaches zero. - The authors therefore study a practical special case in which all jobs share a common deadline. ## A Common-Deadline Algorithm For a unit-capacity system, the algorithm maintains a tentative schedule of jobs in disjoint time intervals. When a new job arrives, it applies the first suitable action: 1. Place the job in an empty interval. 2. Replace a scheduled future job if the new job is significantly shorter. 3. Interrupt the current job if the new job is shorter than its remaining processing time. 4. Discard the new job. - The approach balances immediate execution against preserving capacity for shorter future jobs. - A generalized version works with arbitrary capacity profiles. - The resulting algorithm achieves the first constant competitive guarantee for this setting: **1/11**. The results suggest that schedulers for volatile cloud environments need controlled interruption and carefully designed replacement policies. Allowing restarts offers strong guarantees, while stricter interruption rules require additional structure—such as a shared deadline—to achieve predictable performance.

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Approximation Algorithms Cloud Computing Resource Management Scheduling Algorithms+3

googleJan 23, 20263 min readCurated summary

Introducing GIST: The Next Stage in Smart Sampling | Google Research

GIST is a data-subset selection algorithm designed to balance diversity and utility when training on massive datasets. It converts the difficult diversity–utility optimization problem into a series of thresholded graph problems and uses a bicriteria greedy strategy to find a high-quality subset efficiently. The algorithm guarantees at least half the value of the optimal solution, while the authors prove that improving beyond a 0.56 approximation is NP-hard. ## Why Smart Sampling Is Difficult - Large ML systems need to process datasets that are increasingly expensive to store, analyze, and train on. - Subset selection aims to choose a smaller but representative set of examples. - **Diversity** prevents redundant selections by maximizing the minimum distance between selected points, typically in embedding space. - **Utility** measures how much relevant or unique information the subset provides, modeled using monotone submodular functions. - Optimizing both objectives simultaneously is NP-hard: - A diversity-only method may select irrelevant examples. - A utility-only method may select many similar examples from one highly relevant cluster. ## How GIST Works ### Diversity Thresholding - GIST fixes a candidate minimum distance rather than optimizing the distance directly. - It builds a graph in which two data points are connected when they are closer than the chosen threshold. - Connected points are considered too similar to coexist in the selected subset. - Selecting points that are not connected enforces the desired spacing between examples. ### Utility-Constrained Independent Sets - For each threshold, GIST seeks a high-utility independent set: a group of points with no edges between them. - This corresponds to selecting valuable examples without choosing mutually conflicting or overly similar points. - Because maximum independent set is NP-complete and lacks practical general-purpose approximation algorithms, GIST uses a specially designed bicriteria greedy method. - The algorithm repeatedly selects high-scoring points and excludes nearby candidates, effectively creating “no-go zones” around selected data. ### Searching Across Thresholds - GIST evaluates all relevant distance thresholds derived from the dataset. - It greedily constructs a candidate subset for each threshold. - It returns the best candidate found across these runs. - If the optimal solution achieves minimum distance $d$, GIST obtains comparable utility while guaranteeing a minimum distance of roughly $d/2$. ## Theoretical Guarantees - GIST is presented as the first algorithm with a strong provable guarantee for this diversity–utility tradeoff. - Its output has at least half the value of the absolute optimum. - The authors also prove that finding a solution worth more than 0.56 of the optimum is NP-hard. - These results provide a mathematical guarantee that GIST is not merely producing empirically good subsets, but making a bounded tradeoff between informativeness and coverage. ## Practical Evaluation - GIST was evaluated against several common subset-selection approaches in ML applications. - Comparisons included: - **Random**, a simple baseline that often provides reasonable diversity. - **Margin**, which selects examples the model is uncertain about but does not explicitly promote diversity. - **k-center**, which minimizes representation gaps by keeping all data points close to a selected representative. - **Submod**, which combines utility with an older formulation of diversity. - The experiments, including image-classification benchmarks, reportedly show that GIST outperforms state-of-the-art alternatives while retaining formal guarantees. GIST is therefore a practical choice when subset selection must preserve both broad data coverage and task relevance. Its main advantage is combining competitive real-world performance with a clear approximation guarantee, rather than relying solely on heuristic results.

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Approximation Algorithms Machine Learning Subset Selection Data Sampling+4

googleSep 29, 2025Original article

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.

Approximation Algorithms AI Large Language Models Gemini+5