Automated planning involves finding a sequence of actions that changes the world from an initial state to a final state with goals satisfied. The general problem is PSPACE-hard. Nevertheless, many restricted variants are NP-complete or even in P. Existing complexity work focuses mostly on plan existence, or plan with minimal plan length. Little is known about optimization variants that aim to satisfy as many goal conditions as possible. In this paper, we aim to fill this gap by providing a first inapproximability study of goal-maximization using the classical STRIPS formalism. For MAXPLANSAT and its length-bounded counterpart MAXPLANSAT(K), we prove tight constant-factor lower bounds. More specifically, through performing L-reductions from MAXE3SAT and MAX3DM, we show several of these problems are inapproximable by a constant factor, unless P=NP.
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First-order relational languages have been used in MDP planning and reinforcement learning (RL) for two main purposes: specifying MDPs in compact form, and representing and learning policies that are general and not tied to specific instances or state spaces. In this work, we instead consider the use of first-order languages in goal-conditioned RL and generalized planning. The question is how to learn goal-conditioned and general policies when the training instances are large and the goal cannot be reached by random exploration alone. The technique of Hindsight Experience Replay (HER) provides an answer to this question: it relabels unsuccessful trajectories as successful ones by replacing the original goal with one that was actually achieved. If the target policy must generalize across states and goals, trajectories that do not reach the original goal states can enable more data- and time-efficient learning. In this work, we show that further performance gains can be achieved when states and goals are represented by sets of atoms. We consider three versions: goals as full states, goals as subsets of the original goals, and goals as lifted versions of these subgoals. The result is that the latter two successfully learn general policies on large planning instances with sparse rewards by automatically creating a curriculum of easier goals of increasing complexity. The experiments illustrate the computational gains of these versions, their limitations, and opportunities for addressing them.
Online planning in Markov Decision Processes (MDPs) enables agents to make sequential decisions by simulating future trajectories from the current state, making it well-suited for large-scale or dynamic environments. Sample-based methods such as Sparse Sampling and Monte Carlo Tree Search (MCTS) are widely adopted for their ability to approximate optimal actions using a generative model. However, in practical settings, the generative model is often learned from limited data, introducing approximation errors that can degrade performance or lead to unsafe behaviors. To address these challenges, Robust MDPs (RMDPs) offer a principled framework for planning under model uncertainty, yet existing approaches are typically computationally intensive and not suited for real-time use. In this work, we introduce Robust Sparse Sampling (RSS), the first online planning algorithm for RMDPs with finite-sample theoretical performance guarantees. Unlike Sparse Sampling, which estimates the nominal value function, RSS computes a robust value function by leveraging the efficiency and theoretical properties of Sample Average Approximation (SAA), enabling tractable robust policy computation in online settings. RSS is applicable to infinite or continuous state spaces, and its sample and computational complexities are independent of the state space size. We provide theoretical performance guarantees and empirically show that RSS outperforms standard Sparse Sampling in environments with uncertain dynamics.
Plan verification is the task of checking whether a proposed plan correctly solves a given planning problem. In Hierarchical Task Network (HTN) planning, this verification problem is known to be NP-hard. Existing approaches to HTN plan verification range from SAT encodings to parser-based techniques. However, existing methods do not explicitly exploit the temporal structure inherent in hierarchical decomposition. In this paper, we establish a formal connection between HTN planning and temporal reasoning by showing how decomposition structures can be naturally represented using qualitative constraint networks. Building on this insight, we present a new top-down encoding that transforms the verification of partially ordered task networks into a temporal reasoning problem. We prove the correctness of this encoding and explain how it accounts for both the hierarchical and temporal aspects of HTN plans. By linking HTN plan verification with qualitative temporal reasoning, our approach introduces a principled formal framework for reasoning about complex temporal relationships in hierarchical plans. This connection offers new perspectives for knowledge representation in structured planning domains.
Robust Markov decision processes (r-MDPs) extend MDPs by explicitly modelling epistemic uncertainty about transition dynamics. Learning r-MDPs from interactions with an unknown environment enables the synthesis of robust policies with provable (PAC) guarantees on performance, but this can require a large number of sample interactions. We propose novel methods for solving and learning r-MDPs based on factored state-space representations that leverage the independence between model uncertainty across system components. Although policy synthesis for factored r-MDPs leads to hard, non-convex optimisation problems, we show how to reformulate these into tractable linear programs. Building on these, we also propose methods to learn factored model representations directly. Our experimental results show that exploiting factored structure can yield dimensional gains in sample efficiency, producing more effective robust policies with tighter performance guarantees than state-of-the-art methods.
We address the synthesis of control policies for unknown discrete-time stochastic dynamical systems to satisfy temporal logic objectives. We present a data-driven, abstraction-based control framework that integrates online learning with novel incremental game-solving. Under appropriate continuity assumptions, our method abstracts the system dynamics into a finite stochastic (2.5-player) game graph derived from data. Given a requirement over time on this graph, we compute the winning region -- i.e., the set of initial states from which the objective is satisfiable -- in the resulting game, together with a corresponding control policy. Our main contribution is the construction of abstractions, winning regions and control policies incrementally, as data about the system dynamics accumulates. Concretely, our algorithm refines under- and over-approximations of reachable sets for each state-action pair as new data samples arrive. These refinements induce structural modifications in the game graph abstraction -- such as the addition or removal of nodes and edges -- which in turn modify the winning region. Crucially, we show that these updates are inherently monotonic: under-approximations only grow, over-approximations only shrink, and the winning region only expands. We exploit this monotonicity by defining an objective-induced ranking function on the nodes of the abstract game that increases monotonically as new data samples are incorporated. These ranks underpin our novel incremental game-solving algorithm, which employs customized gadgets (DAG-like subgames) within a rank-lifting algorithm to efficiently update the winning region. Numerical case studies demonstrate significant computational savings compared to the baseline approach, which resolves the entire game from scratch whenever new data samples arrive.
Not Everything Is Permitted: Constrained Cartesian Abstractions for Optimal Classical Planning
PDF ↗Cartesian abstractions can flexibly approximate planning tasks to generate admissible heuristic functions. Constrained abstractions use state constraints, such as mutexes, to eliminate parts of the abstraction that cannot belong to solutions for the original problem. While this has been successfully applied to simple forms of abstraction, no previous work has explored how to do this for Cartesian abstractions. We introduce constrained Cartesian abstractions, which leverage state constraints in multiple ways: to prune spurious transitions and to simplify or even remove abstract states. Moreover, we also use disambiguation to better guide the counterexample-guided process used to generate the abstractions. Our experimental results show that the resulting constrained Cartesian abstractions induce more informed heuristics than their non-constrained counterpart.
Uncertainty over model knowledge is a core challenge in planning and has been addressed through various approaches tailored to different scenarios. In this paper, we focus on scenarios where the agent does not initially know the exact outcome of its actions but gains knowledge upon execution, i.e., each action reveals its actual effect, removing uncertainty about future occurrences. We refer to this formulation as Planning with Uncertain Models of Actions (PUMA). We show that PUMA can be compiled in polynomial time in both Fully Observable Non-Deterministic planning and, perhaps more unexpectedly, classical planning, providing a constructive proof that PUMA remains PSPACE-complete despite its apparent exponential uncertainty. Finally, we experimentally evaluate both compilations with benchmark domains that capture the key aspects of the problem. The results show the practical feasibility of our approach and reveal a complementary behavior between the two compilations.
Modern planning systems utilize various plan representations - sequential, parallel, partially ordered (PO), and partial-order causal link (POCL) - each with different models for concurrency. These formalisms are often implicitly assumed to have the same base properties, particularly regarding makespan. We challenge this assumption, proving the relationship between them is fundamentally asymmetric. Our analysis shows conversions from plans with rigid concurrency layers (sequential, parallel) to those with flexible partial orders (PO, POCL) can preserve makespan. However, the reverse generally fails; the flexible orderings in PO/POCL plans can yield shorter makespans for solutions that cannot be represented in parallel plans without serialization. We prove that finding an optimal parallel representation for a given POCL plan is NP-complete, resolving a key question about their practical interchangeability. We also provide tight complexity bounds for makespan-bounded plan existence. Notably, our results disprove a claim in the literature that planning graph-based planners maximize concurrency by minimizing the critical path in derived PO plans.
We study planning in a fragment of PDDL with qualitative state-trajectory constraints, capturing safety requirements, task ordering conditions, and intermediate sub-goals commonly found in real-world problems. A prominent approach to tackle such problems is to compile their constraints away, leading to a problem that is supported by state-of-the-art planners. Unfortunately, existing compilers do not scale on problems with a large number of objects and high-arity actions, as they necessitate grounding the problem before compilation. To address this issue, we propose two methods for compiling away constraints without grounding, making them suitable for large-scale planning problems. We prove the correctness of our compilers and outline their worst-case time complexity. Moreover, we present a reproducible empirical evaluation on the domains used in the latest International Planning Competition. Our results demonstrate that our methods are efficient and produce planning specifications that are orders of magnitude more succinct than the ones produced by compilers that ground the domain, while remaining competitive when used for planning with a state-of-the-art planner.
Automated Repair of Totally-Ordered Hierarchical Task Network Domains via Context-Free Grammars with Large Language Model Support
PDF ↗Repairing flawed domain models remains a critical challenge in AI planning, with few effective techniques available. We propose a novel approach for repairing totally ordered hierarchical task network (TO-HTN) models with missing actions, guided by a plan that must be valid for the repaired model. This problem has only one previously documented approach, which relies on complex re-encoding that's solved via TO-HTN planning. In contrast, our approach translates the repair task into a context-free grammar repair problem and leverages a large language model (LLM) to identify and insert relevant actions directly, simplifying the repair process. We evaluate our approach on established benchmarks and demonstrate substantially improved results over the prior approach, achieving nearly three times the number of instances solved, and nearly solving all instances of domains in which the previous approach solved zero. Importantly, we mask all natural language hints, such as action names, forcing the LLM to simulate reasoning and planning, and mitigating the risk of data leakage from its training corpus.
Realistic background traffic is critical to the simulation platforms for autonomous driving (AD) testing. Given that most vehicles in reality are driven by human beings, introducing human driving (HD) vehicles to the background traffic is necessary to be able to discover more problems of the tested AD vehicle in the simulation stage. However, existing methods rely on ad-hoc rules or data-driven training to mimic partial human driver behaviors, which are not comprehensive and lack transparency. In this work, we design a smart human driving vehicle simulator HDSim which is empowered by cognitively inspired modeling and AI models. HDSim enables diverse, realistic, and scalable HD traffic simulation on AD testing platforms like CARLA in a non-intrusive manner. There are two novel components in HDSim. First, we introduce a driver model to guide the generation of diverse human driving styles by using different combinations of latent cognitive factors in a hierarchy. Second, we design a Perception-Mediated Behavior Influence (PMBI) mechanism to use LLM-assisted perceptual transformations to indirectly fuse driving actions with driving styles. Experiments show that HDSim traffic can help simulation platforms like CARLA to reveal 68% more failures of tested AD vehicles, and the explainability of reported accidents is also improved.
Machine learning methods have been increasingly applied to solve Vehicle Routing Problems (VRPs). A high-efficiency approach is to learn solution construction using deep neural networks. However, their tendency toward premature convergence is a critical barrier, severely hindering generalization across diverse distributions and scales. To overcome this, we introduce Elite-Pattern Reinforcement (EPR), a novel strategy designed to create a synergy between the diverse, exploratory nature of reinforcement learning and the high-quality, structured knowledge from classical heuristics. The strategy guides the learning process by reinforcing structural patterns from elite solutions, employing an elite-guided score modulation to integrate this external knowledge. The inherent symmetry of path patterns is also exploited to augment the structural information. This steers the policy away from premature convergence by enabling it to distinguish and favour elite path patterns over inferior ones. Integrating our strategy with four construction methods yields substantial performance improvements on the CVRPLIB and TSPLIB benchmarks. Furthermore, our approach outperforms state-of-the-art learning-based methods, demonstrating superior generalization across diverse distributions and scales.
Boosting Cross-problem Generalization in Diffusion-Based Neural Combinatorial Solver via Inference Time Adaptation
PDF ↗Diffusion-based Neural Combinatorial Optimization (NCO) has demonstrated effectiveness in solving NP-complete (NPC) problems by learning discrete diffusion models for solution generation, eliminating hand-crafted domain knowledge. Despite their success, existing NCO methods face significant challenges in both cross-scale and cross-problem generalization, and high training costs compared to traditional solvers. While recent studies on diffusion models have introduced training-free guidance approaches that leverage pre-defined guidance functions for conditional generation, such methodologies have not been extensively explored in combinatorial optimization. To bridge this gap, we propose a training-free inference time adaptation framework (DIFU-Ada) that enables both the zero-shot cross-problem transfer and cross-scale generalization capabilities of diffusion-based NCO solvers without requiring additional training. We provide theoretical analysis that helps understanding the cross-problem transfer capability. Our experimental results demonstrate that a diffusion solver, trained exclusively on the Traveling Salesman Problem (TSP), can achieve competitive zero-shot transfer performance across different problem scales on TSP variants, such as Prize Collecting TSP (PCTSP) and the Orienteering Problem (OP), through inference time adaptation.
Hierarchical Reinforcement Learning with Topology-Aware Exploration Framework for Multi-path Commodity Flow Problem
PDF ↗The multi-path commodity flow problem (MPCFP) is crucial for ensuring reliable and high-speed data transmission in communication networks. However, existing studies that employ pre-generated routing paths neglect real-time load state and the coupling among decisions, thus hindering the achievement of high-quality solutions. To overcome this, we propose Hierarchical Reinforcement Learning with Topology-Aware Exploration (HRL-TAE), which is the first fully end-to-end framework that dynamically produces high-quality solutions based on real-time network states. HRL-TAE integrates an exploration mechanism and utilizes the State Transition Guiding List (STGL) to guide state transitions, thereby transforming topology exploration into a Markov decision process. Guided by STGL, two closely coupled layers in HRL-TAE, that is, the path construct layer and the ratio allocate layer, construct multiple subpaths for each flow and allocate traffic ratios among them. Subsequently, adaptive constraint-driven masks exclude infeasible actions during decision making, thereby guaranteeing that all constraints are satisfied. We also adopt a tailored training approach to obtain accurate gradient estimates and improve training efficiency. Simulations and real-world experiments demonstrate that HRL-TAE achieves superior performance.
Experiential Fairness: Bridging the Gap Between User Experience and Resource-Centric Fairness in Online LLM Services
PDF ↗Conventional fairness in multi-tenant Large Language Model (LLM) inference services is typically defined by system-centric metrics such as equitable resource allocation. We argue that this is unilateral and it creates a gap between measured system performance and actual user-perceived quality. We challenge this notion by introducing and formalizing Experiential Fairness, a user-centric paradigm that shifts the objective from equality of opportunity (resource access) to equity of outcome (user experience). With this motivation we propose ExFairS, a lightweight scheduling framework that perceives each user's satisfaction as a composite measure of Service Level Objective (SLO) compliance and resource consumption, and dynamically re-orders the serving queue guided by a credit-based priority mechanism. Extensive experiments on an 8-GPU NVIDIA V100 node show that ExFairS reduces the SLO violation rate by up to 100% and improves system throughput by 14-21.9%, outperforming state-of-the-art schedulers and delivering a demonstrably higher degree of Experiential Fairness.
In recent years, ML-based heuristic functions for automated planning have shown increasing performance. A main challenge is the level of generalization required in planning: techniques must generalize at least across different instances of the same domain (which results in different sizes of learning input). A common approach to overcome the issue is to use graph representations as input. While GNNs are a natural choice for learning, other methods have recently been favored because they show better runtime performance and need less training data. However, existing work has so far been limited to non-hierarchical planning. We describe the first approach to learn heuristics for hierarchical planning. We extend the Instance Learning Graph – a graph structure used in non-hierarchical planning – to the new setting and show how to learn heuristic functions based on it. Since our heuristics are applicable to the lifted model, there is no need to ground it. We therefore combine it with a novel lifted HTN planning system. Like recent systems in non-hierarchical planning, it grounds the search space explored so far, but not the entire model prior to search. Our evaluation shows that our approach is competitive with the lifted systems from the literature, though the ground systems achieve higher coverage.
Constrained and Robust Policy Synthesis with Satisfiability-Modulo-Probabilistic-Model-Checking
PDF ↗The ability to compute reward-optimal policies for given and known finite Markov decision processes (MDPs) underpins a variety of applications across planning, controller synthesis, and verification. However, we often want policies (1) to be robust, i.e., they perform well on perturbations of the MDP and (2) to satisfy additional structural constraints regarding, e.g., their representation or implementation cost. Computing such robust and constrained policies is indeed computationally more challenging. This paper contributes the first approach to effectively compute robust policies subject to arbitrary structural constraints using a flexible and efficient framework. We achieve flexibility by allowing to express our constraints in a first-order theory over a set of MDPs, while the root for our efficiency lies in the tight integration of satisfiability solvers to handle the combinatorial nature of the problem and probabilistic model checking algorithms to handle the analysis of MDPs. Experiments on a few hundred benchmarks demonstrate the feasibility for constrained and robust policy synthesis and the competitiveness with state-of-the-art methods for various fragments of the problem.
This paper proposes a framework for improving the operational efficiency of automated storage systems under uncertainty. It considers a 2D grid-based storage for uniform-sized loads (e.g., containers, pallets, or totes), which are moved by a robot (or other manipulator) along a collision-free path in the grid. The loads are labeled (i.e., unique) and must be stored in a given sequence, and later be retrieved in a different sequence---an operational pattern that arises in logistics applications, such as last-mile distribution centers and shipyards. The objective is to minimize the load relocations to ensure efficient retrieval. A previous result guarantees a zero-relocation solution for known storage and retrieval sequences, even for storage at full capacity, provided that the side of the grid through which loads are stored/retrieved is at least 3 cells wide. However, in practice, the retrieval sequence can change after the storage phase. To address such uncertainty, this work investigates k-bounded perturbations during retrieval, under which any two loads may depart out of order if they are originally at most k positions apart. We prove that a Theta(k) grid width is necessary and sufficient for eliminating relocations at maximum capacity. We also provide an efficient solver for computing a storage arrangement that is robust to such perturbations. To address the higher-uncertainty case where perturbations exceed k, a strategy is introduced to effectively minimize relocations. Extensive experiments show that, for k up to half the grid width, the proposed storage-retrieval framework essentially eliminates relocations. For k values up to the full grid width, relocations are reduced by 50%+.
While transformers excel in many settings, their application in the field of automated planning is limited. Prior work like PlanGPT, a state-of-the-art decoder-only transformer, struggles with extrapolation from easy to hard planning problems. This in turn stems from problem symmetries: planning tasks can be represented with arbitrary variable names that carry no meaning beyond being identifiers. This causes a combinatorial explosion of equivalent representations that pure transformers cannot efficiently learn from. We propose a novel contrastive learning objective to make transformers symmetry-aware and thereby compensate for their lack of inductive bias. Combining this with architectural improvements, we show that transformers can be efficiently trained for either plan-generation or heuristic-prediction. Our results across multiple planning domains demonstrate that our symmetry-aware training effectively and efficiently addresses the limitations of PlanGPT.