Lazy clause generation (LCG) is an approach to building constraint programming (CP) solvers that uses ideas from Boolean Satisfiability (SAT) solvers. Propagation in the CP solver is extended to explain itself, lazily creating explanation clauses. This allows SAT style conflict analysis to be used to create clausal nogoods that record the reason for failure, and avoid the same failure in the future. LCG has proven to be a highly effective architecture, used in all most performant CP solvers. In this review we consider how LCG has been refined and expanded over the years since its invention.

Modern combinatorial solvers can be understood as searching for proofs of unsatisfiability or optimality. The proof system implemented by a solver, therefore, fundamentally shapes solver performance. While propositional resolution is simple and complete for propositional formulas, no existing practical resolution-based system offers unrestricted complete reasoning over integer linear inequalities. We introduce hypercube linear resolution, a new proof system that is both sound and complete for integer linear reasoning. Hypercube linear resolution integrates propositional resolution with Fourier resolution through a new constraint type, the hypercube linear constraint, which captures linear relations within a discrete hypercube. Our main contribution is the theory of the proof system, establishing its soundness and completeness. Our proof system generalises propositional and Fourier resolution, and can be seen as an extended cutting planes proof system. We also provide a conflict-driven search algorithm that exploits the system in practice. Our preliminary experiments demonstrate that the new system reduces conflicts compared to propositional resolution. This shows that the structure captured by hypercube linear constraints can be exploited to improve constraint solving.

Recent vision-language models have strong perceptual ability but their implicit reasoning is hard to explain and easily generates hallucinations on complex queries. Compositional methods improve interpretability, but most rely on a single agent or hand-crafted pipeline and cannot decide when to collaborate across complementary agents or compete among overlapping ones. We introduce MATA (Multi-Agent hierarchical Trainable Automaton), a multi-agent system presented as hierarchical finite-state automaton for visual reasoning whose top-level transitions are chosen by a trainable hyper agent. Each agent corresponds to a state in the hyper automaton, and runs a small rule-based sub-automaton for reliable micro-control. All agents read and write a shared memory, yielding transparent execution history. To supervise the hyper agent’s transition policy, we build transition-trajectory trees and transform to memory-to-next-state pairs, forming the MATA-SFT-90K dataset for supervised finetuning (SFT). The finetuned LLM as the transition policy understands the query and the capacity of agents, and it can efficiently choose the optimal agent to solve the task. Across multiple visual reasoning benchmarks, MATA achieves the state-of-the-art results compared with monolithic and compositional baselines.

Single transferable vote (STV) is a multi-winner preferential proportional electoral system. The margin is the smallest number of ballots that need to be manipulated to alter the set of winners. If we can compute the margin of an STV election, or a reasonable lower bound on the margin, we can use recent advances in auditing research to conduct a risk-limiting audit of the election's winners. Knowledge of the margin also provides insight into whether uncovered mistakes, or a known error rate in ballot interpretation, could have influenced the outcome. This paper presents substantial improvements on an existing algorithm for computing lower bounds on the margin of an STV election. These improvements allow us to compute higher lower bounds for real STV elections, making mismatch-based risk-limiting audits more practical.

Difference constraints of the form x - y ≤ d are well studied, with efficient algorithms for satisfaction and implication, because of their connection to shortest paths. Finite domain propagation algorithms, however, typically do not make use of these algorithms, and treat each difference constraint as a separate propagator. Propagation does guarantee completeness of solving, but can be needlessly slow. In this paper we describe how to build a (bounds consistent) global propagator for difference constraints that treats them all simultaneously. SAT modulo theory solvers have included theory solvers for difference constraints for some time. While a theory solver for difference constraints gives the basis of a global difference constraint propagator, we show how the requirements on the propagator are quite different. Crucially, we show how to explain propagations by a global difference constraint propagator, in order to use it within a lazy clause generation solver. We give experiments showing that treating difference constraints globally can substantially improve on the standard propagation approach.

Visual Grounding (VG) tasks, such as referring expression detection and segmentation tasks are important for linking visual entities to context, especially in complex reasoning tasks that require detailed query interpretation. This paper explores VG beyond basic perception, highlighting challenges for methods that require reasoning like human cognition. Recent advances in large language methods (LLMs) and Vision-Language methods (VLMs) have improved abilities for visual comprehension, contextual understanding, and reasoning. These methods are mainly split into end-to-end and compositional methods, with the latter offering more flexibility. Compositional approaches that integrate LLMs and foundation models show promising performance but still struggle with complex reasoning with language-based logical representations. To address these limitations, we propose NAVER, a compositional visual grounding method that integrates explicit probabilistic logic reasoning within a finite-state automaton, equipped with a self-correcting mechanism. This design improves robustness and interpretability in inference through explicit logic reasoning. Our results show that NAVER achieves SoTA performance comparing to recent end-to-end and compositional baselines. The code is available at https://github.com/ControlNet/NAVER .

In multi-agent systems, when we account for the possibility of delays during execution, online planning becomes more complicated, as both execution and planning should be able to handle delays when agents are moving. Lifelong Multi-Agent Path Finding (LMAPF) is the problem of (re)planning the collision-free moves of agents to their goals in a shared space, while agents continuously receive new goals. PIE (Planning and Improving while Executing) is a recent approach to LMAPF which concurrently replans later parts of agents' trajectories while execution occurs. However, the execution is assumed to be perfect. Existing approaches either use policy-based methods to quickly coordinate agents every timestep with instant delay feedback, or deploy an execution policy to adjust a solution for delays on the fly. These approaches may introduce large amounts of unnecessary delays to agents due to their planner guarantees or simple delay-handling policies. In this paper, we extend PIE to define a framework for solving the lifelong MAPF problem with execution delays. We instantiate our framework with different execution and replanning strategies, and experimentally evaluate them. Overall, we find that this framework can substantially improve the throughput by up to a factor 3 for lifelong MAPF, compared to approaches that handle delays with simple execution policies.

Explainable Artificial Intelligence (XAI) is critical for attaining trust in the operation of AI systems. A key question of an AI system is ``why was this decision made this way''. Formal approaches to XAI use a formal model of the AI system to identify abductive explanations. While abductive explanations may be applicable to a large number of inputs sharing the same concrete values, more general explanations may be preferred for numeric inputs. So-called inflated abductive explanations give intervals for each feature ensuring that any input whose values fall withing these intervals is still guaranteed to make the same prediction. Inflated explanations cover a larger portion of the input space, and hence are deemed more general explanations. But there can be many (inflated) abductive explanations for an instance. Which is the best? In this paper, we show how to find a most general abductive explanation for an AI decision. This explanation covers as much of the input space as possible, while still being a correct formal explanation of the model's behaviour. Given that we only want to give a human one explanation for a decision, the most general explanation gives us the explanation with the broadest applicability, and hence the one most likely to seem sensible.

Constructing efficient risk-limiting audits (RLAs) for multiwinner single transferable vote (STV) elections is a challenging problem. An STV RLA is designed to statistically verify that the reported winners of an election did indeed win according to the voters' expressed preferences and not due to mistabulation or interference, while limiting the risk of accepting an incorrect outcome to a desired threshold (the risk limit). Existing methods have shown that it is possible to form RLAs for two-seat STV elections in the context where the first seat has been awarded to a candidate in the first round of tabulation. This is called the first winner criterion. We present an assertion-based approach to conducting full or partial RLAs for STV elections with three or more seats, in which the first winner criterion is satisfied. Although the chance of forming a full audit that verifies all winners drops substantially as the number of seats increases, we show that we can quite often form partial audits that verify most, and sometimes all, of the reported winners. We evaluate our method on a dataset of over 500 three- and four-seat STV elections from the 2017 and 2022 local council elections in Scotland.

In this paper we focus on a challenging version of the aircraft maintenance routing problem (AMRP) with a maintenance distribution objective (AMRP-D). For the AMRP-D, the flight legs with predefined start and end times are assigned to aircraft. In addition to the assigned flight legs, each aircraft has to satisfy certain regulations regarding the maintenance services that are mandatory in the scheduling period, while the maintenance should also be distributed evenly. We propose a two stage approach, where first we use a decomposition method that is solved using constraint programming, to cover the flight legs. To optimize the distribution objective, we propose two metaheuristic techniques based on Large Neighborhood Search (LNS) and Simulated Annealing (SA). The LNS method consists of different destroy operators and as repairer we use a constraint programming (CP) solver. The SA approach includes a novel neighborhood to deal with the distribution objective. Our experimental results show that the decomposition method is able to solve more instances than the exact approach, while SA provides better quality solutions for the optimization stage compared to LNS.

One approach to risk-limiting audits (RLAs) compares randomly selected cast vote records (CVRs) to votes read by human auditors from the corresponding ballot cards. Historically, such methods reduce audit sample sizes by considering how each sampled CVR differs from the corresponding true vote, not merely whether they differ. Here we investigate the latter approach, auditing by testing whether the total number of mismatches in the full set of CVRs exceeds the minimum number of CVR errors required for the reported outcome to be wrong (the "CVR margin"). This strategy makes it possible to audit more social choice functions and simplifies RLAs conceptually, which makes it easier to explain than some other RLA approaches. The cost is larger sample sizes. "Mismatch-based RLAs" only require a lower bound on the CVR margin, which for some social choice functions is easier to calculate than the effect of particular errors. When the population rate of mismatches is low and the lower bound on the CVR margin is close to the true CVR margin, the increase in sample size is small. However, the increase may be very large when errors include errors that, if corrected, would widen the CVR margin rather than narrow it; errors affect the margin between candidates other than the reported winner with the fewest votes and the reported loser with the most votes; or errors that affect different margins.

Tree Cache is a lightweight pre-processing approach to grid path finding which works by generating a shortest path tree: from a root cell to all cells in the map. During online search Tree Cache simply follows the tree: from start and target towards the root, stopping at the first common cell. Although Tree Cache is fast, the resulting paths have no solution quality guarantees. In this paper we improve Tree Cache, in terms of speed and solution quality, by combining symmetry breaking ideas from Jump Point Search. Our new algorithm, Jump Spanning Tree Search (JSTS), can usually generate paths with low average sub-optimality in under one microsecond -- up to two orders of magnitude faster than Tree Cache. We then extend JSTS to derive a new and very fast bounded suboptimal search, which guarantees solution quality in single-digit microseconds on average. Our results establish a remarkable new level of performance in the area. In particular, we show JSTS approaches and often improves upon the output complexity of an idealised oracle, which simply reads off and returns a corresponding but optimal solution path.

Discrete optimization models often reason about discrete sets of objects, but discrete optimization solvers only deal with integers. One of the key challenges when building models for discrete optimization problems is avoiding bugs. Because the model only defines constraints, decisions, and an objective that are then run on a solver, bugs in the model can be very difficult to track down. Hence, modelling languages should have strong type systems to detect as many bugs as possible at the modelling level. In this paper, we propose unit types for MiniZinc. Unit types allow us to differentiate between different integers appearing in the model. Almost all integer decisions in models are either about a set of objects or some measurable resource type. Using unit types, we can add more type safety to our models by avoiding confusion of decisions on different resource types. Compared to other programming languages, unit types in our proposal are unusual. MiniZinc models often deal with multiple levels of granularity of the same resource, e.g., scheduling to the minute, but doing resource allocation on the half day, or use an unspecified granularity, e.g., the same job-shop scheduling model could use task durations given in minutes or days. Our proposed unit types also differentiate between coordinate unit types, e.g., the time when an event occurred, and the usual delta unit types, e.g., the time difference between two events. Errors arising from mixing coordinate and delta types can be very challenging to debug, so we extend the type system to track this for us.

We investigate the effectiveness of splitting the search space in parallelising the state-of-the-art CP-SAT solver. One of the key barriers to effective search-space splitting in learning solvers is the generated sub-problems are not independent, leading to substantial communication-related overhead, substantial redundant work, or both. Our contributions attempt to mitigate this issue: job reassignment; and trail sharing. Jobs (sub-trees) are reassigned to new workers if the clause database of the currently assigned worker appears ill-suited to the region of the search-space, when doing so workers can share some of the state from their local trail. We argue a trail prefix can be viewed as a lossy compressed representation of much of the relevant information a worker has learnt about a job, this information can be exploited by the next worker assigned the same subtree to avoid some redundant work. We show these approaches complement standard portfolio and clause-sharing approaches, improving CP-SAT's performance on MiniZinc challenge benchmarks with a moderate number of worker threads. To enable these approaches, we also introduce "Buffered Work Stealing," which can be parameterised to emulate the two main existing approaches to search-space splitting in the literature: Work Stealing and Embarassingly Parallel Search, as well as an intermediate configuration between these two extremes that slightly outperforms both.

Single Transferable Vote (STV) counting, used in several jurisdictions in Australia, is a system for choosing multiple election winners given voters' preferences among candidates. The system is complex and it is not always obvious how an individual's vote contributes to candidates' tallies across rounds of tabulation. This short paper presents a demonstration system that allows voters to enter an example vote in a past Australian STV election, and see: (i) how that vote would have been transferred between candidates; and (ii) how much that vote would have contributed to the tallies of relevant candidates, across rounds of tabulation.

The single transferable vote (STV) is a system of preferential proportional voting employed in multi-seat elections. Each ballot cast by a voter is a (potentially partial) ranking over a set of candidates. The margin of victory, or simply margin, is the smallest number of ballots that, if manipulated (e.g., their rankings changed, or ballots being deleted or added), can alter the set of winners. Knowledge of the margin of an election gives greater insight into both how much time and money should be spent on auditing the election, and whether uncovered mistakes (such as ballot box losses) throw the election result into doubt – requiring a costly repeat election – or can be safely ignored. Lower bounds on the margin can also be used for this purpose, in cases where exact margins are difficult to compute. There is one existing approach to computing lower bounds on the margin of STV elections, while there are multiple approaches to finding upper bounds. In this paper, we present improvements to this existing lower bound computation method for STV margins. In many cases the improvements compute tighter (higher) lower bounds as well as making the computation of lower bounds more computationally efficient. For small elections, in conjunction with existing upper bounding approaches, the new algorithms are able to compute exact margins of victory.

We study the problem of online Multi-Agent Pickup and Delivery (MAPD), where a team of agents must repeatedly serve dynamically appearing tasks on a shared map. Existing online methods either rely on simple heuristics, which result in poor decisions, or employ complex reasoning, which suffers from limited scalability under real-time constraints. In this work, we focus on the task assignment subproblem and formulate it as a minimum-cost flow over the environment graph. This eliminates the need for pairwise distance computations and allows agents to be simultaneously assigned to tasks and routed toward them. The resulting flow network also supports efficient guide path extraction to integrate with the planner and accelerates planning under real-time constraints. To improve solution quality, we introduce two congestion-aware edge cost models that incorporate real-time traffic estimates. This approach supports real-time execution and scales to over 20000 agents and 30000 tasks within 1-second planning time, outperforming existing baselines in both computational efficiency and assignment quality.