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Yujiao Zhao, Yizhan Xiang, Jiangnan Li, Yiyuan Wang, Minghao Yin

Pseudo-Boolean optimization (PBO) problem involves optimizing a linear objective function under linear inequality constraints defined over Boolean variables. PBO is widely used for modeling many combinational optimization problems, particularly in some real-world scenarios. In core-guided CDCL-based exact solvers, the way branching variables are assigned, known as phase selection, significantly affects the solving efficiency. This paper introduces two strategies to enhance solver performance by improving phase selection. Firstly, we design a new phase selection strategy that actively guides variables in the objective function toward assignments closer to the optimal solution. Secondly, to prevent the solver from becoming trapped in local solutions, we propose a reinforcement learning-based rephase mechanism that dynamically updates and resets variable phases. We integrate two phase selection strategies into two state-of-the-art PBO solvers and compare them against top-performing solvers from the PB competitions, using benchmarks from these competitions for assessment. The experimental results show that our solvers outperform the winning solver from the competitions.

Yang Zhang, Hongbo Li

Failure-based variable ordering heuristics (VOH) are efficient general-purpose search heuristics for solving constraint satisfaction problems (CSP). They learn from the failures detected during the search and select the variables that are most likely to fail. The current failure-based VOHs, i.e. the failure-rate-based (FRBA) and failure-length-based (FLBA), focus on only the failures detected in left branches. In this paper, we investigate how the failure information from right branches affects the performance of the failure-based VOHs. Four strategies utilizing the failure information of right branches are proposed to refine the failure-based VOHs. Our experiments performed with the benchmark instances used in the recent MiniZinc challenges show that utilizing the failures detected in right branches enhances the performance of the failure-based VOHs. The refined version combining all the proposed strategies generally gets the best performance. It demonstrates remarkable superiority over several general-purpose VOHs, including activity-based search, conflict-history search, refined weighted degree, pick/dom, and the existing FRBA, which are considered state-of-the-art. Our study demonstrates that right branches matter in failure-based VOHs.

Shuli Zeng, Sijia Zhang, Feng Wu, Shaojie Tang, Xiangyang Li

Embedding deep neural networks (NNs) into mixed-integer programs (MIPs) is attractive for decision making with learned constraints, yet state-of-the-art monolithic linearisations blow up in size and quickly become intractable. In this paper, we introduce a novel dual-decomposition framework that relaxes the single coupling equality u=x with an augmented Lagrange multiplier and splits the problem into a vanilla MIP and a constrained NN block. Each part is tackled by the solver that suits it best-branch and cut for the MIP subproblem, first-order optimisation for the NN subproblem, so the model remains modular, the number of integer variables never grows with network depth, and the per-iteration cost scales only linearly with the NN size. On the public SurrogateLIB benchmark, our method proves scalable, modular, and adaptable: it runs 120x faster than an exact Big-M formulation on the largest test case; the NN sub-solver can be swapped from a log-barrier interior step to a projected-gradient routine with no code changes; and swapping the MLP for an LSTM backbone still completes the full optimisation in 47s without any bespoke adaptation.

Kaijie Xu, Fandi Meng, Clark Verbrugge, Simon Mark Lucas

In Social Deduction Games (SDGs) such as Avalon, Mafia, and Werewolf, players conceal their identities and deliberately mislead others, making hidden-role inference a central and demanding task. Accurate role identification, which forms the basis of an agent's belief state, is therefore the keystone for both human and AI performance. We introduce CSP4SDG, a probabilistic, constraint–satisfaction framework that analyses gameplay objectively. Game events and dialogue are mapped to four linguistically agnostic constraint classes—evidence, phenomena, assertions, and hypotheses. Hard constraints prune impossible role assignments, while weighted soft constraints score the remainder; information-gain weighting links each hypothesis to its expected value under entropy reduction, and a simple closed-form scoring rule guarantees that truthful assertions converge to classical hard logic with minimum error. The resulting posterior over roles is fully interpretable and updates in real time. Experiments on three public datasets show that CSP4SDG (i) outperforms LLM-based baselines in every inference scenario, and (ii) boosts LLMs when supplied as an auxiliary "reasoning tool." Our study validates that principled probabilistic reasoning with information theory is a scalable alternative—or complement—to heavy-weight neural models for SDGs.

Chucheng Xiang, Ruchao Bao, Biyin Feng, Wenzheng Wu, Zhongyuan Liu, Yirui Guan, Ligang Liu

We present a novel framework for automated interior design that combines large language models (LLMs) with grid-based integer programming to jointly optimize room layout and furniture placement. Given a textual prompt, the LLM-driven agent workflow extracts structured design constraints related to room configurations and furniture arrangements. These constraints are encoded into a unified grid-based representation inspired by ``Modulor". Our formulation accounts for key design requirements, including corridor connectivity, room accessibility, spatial exclusivity, and user-specified preferences. To improve computational efficiency, we adopt a coarse-to-fine optimization strategy that begins with a low-resolution grid to solve a simplified problem and guides the solution at the full resolution. Experimental results across diverse scenarios demonstrate that our joint optimization approach significantly outperforms existing two-stage design pipelines in solution quality, and achieves notable computational efficiency through the coarse-to-fine strategy.

Haoze Wu, Clark Barrett, Nina Narodytska

We are exploring the problem of building an automated reasoning procedure that adaptively tunes the high-level solving strategy for a given problem. There are two main distinctive characteristics of our approach: tuning is performed solely online, unlike the common use of tuning as an offline process; and tuning data comes exclusively from the given instance, so we do not rely on the availability of similar benchmarks and can work with unique challenging instances. Our approach builds on top of the divide-and-conquer paradigm that naturally serves partitioned sub-problems for an automated tuning algorithm to obtain a good solving strategy. We demonstrate performance improvement on two classes of important problems--SAT-solving and neural network verification--and show that our method can learn unconventional solving strategies in some cases.

Ruobing Wang, Xin Li, Yangchuan Wang, Zijian Zhang, Mingzhong Wang

Branch-and-bound (B&B) is a fundamental algorithmic framework for solving Mixed-Integer Linear Programming (MILP) problems, where branching decisions critically affect solver efficiency. Recent learning-based methods apply imitation learning to select branching variables, but their deterministic predictions limit exploration and generalization. In this paper, we propose a novel framework that formulates branching variable selection as a conditional generative process, exploring deep-level decision features. Our approach leverages diffusion models to enable diverse and exploratory branching score generation, while consistency modeling distills this process into efficient one-step inference conditioned on the B&B state. This mode allows our method to achieve both high-quality and fast branching decisions, significantly improving the overall performance of branch-and-bound solvers. Extensive experiments on challenging cross-scale and cross-category benchmarks demonstrate that our framework consistently outperforms state-of-the-art imitation learning baselines, delivering substantial improvements in solution quality, computational efficiency, and inference speed.

Dieter Vandesande, Jordi Coll, Bart Bogaerts

Over the past few decades, combinatorial solvers have seen remarkable performance improvements, enabling their practical use in real-world applications. In some of these applications, ensuring the correctness of the solver's output is critical. However, the complexity of modern solvers makes them susceptible to bugs in their source code. In the domain of satisfiability checking (SAT), this issue has been addressed through proof logging, where the solver generates a formal proof of the correctness of its answer. For more expressive problems like MaxSAT, the optimization variant of SAT, proof logging had not seen a comparable breakthrough until recently. In this paper, we show how to achieve proof logging for state-of-the-art techniques in Branch-and-Bound MaxSAT solving. This includes certifying look-ahead methods used in such algorithms as well as advanced clausal encodings of pseudo-Boolean constraints based on so-called Multi-Valued Decision Diagrams (MDDs). We implement these ideas in MaxCDCL, the dominant branch-and-bound solver, and experimentally demonstrate that proof logging is feasible with limited overhead, while proof checking remains a challenge.

Pierre Talbot

Machine learning has tremendously benefited from graphics processing units (GPUs) to accelerate training and inference by several orders of magnitude. However, this success has not been replicated in general and exact combinatorial optimization. Our key contribution is to propose a general-purpose discrete constraint programming solver fully implemented on GPU. It is based on integer interval bound propagation and backtracking search. The two main ingredients are (1) ternary constraint network optimized for GPU architectures, and (2) an on-demand subproblems generation strategy. Our constraint solving algorithm is significantly simpler than those found in optimized CPU constraint solvers, yet is competitive with sequential solvers in the MiniZinc 2024 challenge.

Guidio Sewa, David Allouche, Simon De Givry, George Katsirelos, Pierre Montalbano, Thomas Schiex

To efficiently solve exact discrete optimization problems, branch and bound algorithms require tight bounds. In constraint programming, for optimization, soft arc consistencies typically derive much stronger bounds than those offered by domain or bound consistencies applied to a cost variable. The reason is that soft local consistencies exchange marginal cost information between variables whereas domain consistencies rely only on shrinking domains, which is less informative. However, CP solvers equipped with soft arc consistencies have so far offered limited support for efficient processing of global constraints. In this work, we show how we can efficiently enforce soft local consistency over the AllDifferent constraint, relying on algorithms for the Linear Assignment Problem (LAP). We implement this propagator in toulbar2, the state-of-the-art weighted CP solver exploiting soft local consistencies for bounding. We show that, equipped with this new propagator, toulbar2 outperforms state-of-the-art domain consistency-based CP as well as integer programming solvers for the Quadratic Assignment Problem and shows better performance for miniCOP instances of the 2024 XCSP competition with AllDifferent constraints.

David Saikali, Gilles Pesant

Drug discovery is a very time-consuming and costly endeavour due to its huge design space and to the lengthy and failure-fraught process of bringing a product to market. Automating the generation of candidate molecules exhibiting some of the desired properties can help. Among the standard formats to encode molecules, SMILES is a widespread string representation. We propose a constraint programming model showcasing the grammar constraint to express the design space of organic molecules using the SMILES notation. We show how some common physicochemical properties --- such as molecular weight and lipophilicity --- and structural features can be expressed as constraints in the model. We also contribute a weighted counting algorithm for the grammar constraint, allowing us to use a belief propagation heuristic to guide the generation. Our experiments indicate that such a heuristic is key to driving the search towards desired molecules.

Kunhang Lv, Yuhang Dong, Rui Han, Fuqi Jia, Feifei Ma, Jian Zhang

Quantified formulas with Uninterpreted Functions (UFs) over non-linear real arithmetic pose fundamental challenges for Satisfiability Modulo Theories (SMT) solving. Traditional quantifier instantiation methods struggle because they lack semantic understanding of UF constraints, forcing them to search through unbounded solution spaces with limited guidance. We present AquaForte, a framework that leverages Large Language Models to provide semantic guidance for UF instantiation by generating instantiated candidates for function definitions that satisfy the constraints, thereby significantly reducing the search space and complexity for solvers. Our approach preprocesses formulas through constraint separation, uses structured prompts to extract mathematical reasoning from LLMs, and integrates the results with traditional SMT algorithms through adaptive instantiation. AquaForte maintains soundness through systematic validation: LLM-guided instantiations yielding SAT solve the original problem, while UNSAT results generate exclusion clauses for iterative refinement. Completeness is preserved by fallback to traditional solvers augmented with learned constraints. Experimental evaluation on SMT-COMP benchmarks demonstrates that AquaForte solves numerous instances where state-of-the-art solvers like Z3 and CVC5 timeout, with particular effectiveness on satisfiable formulas. Our work shows that LLMs can provide valuable mathematical intuition for symbolic reasoning, establishing a new paradigm for SMT constraint solving.

Suyu Liu, Zhiguang Cao, Nan Yin, Yew-Soon Ong

Neural solvers for Vehicle Routing Problems (VRPs) have shown great advantages in solving various kinds of problem types. However, they also face critical challenges in generalizing from small-scale training to large-scale problems and in identifying the most salient topological information for decision-making. To mitigate these gaps, we introduce ScaleNet, a novel hierarchical framework that integrates a U-Net architecture into a unified, multi-task VRP solver. Scale-Net explicitly captures multi-scale structural patterns by processing a nested hierarchy of input graph instances. This enriched, coarse-to-fine representation is extracted by the encoder and fed directly into the decoder, empowering decoder module with superior topological awareness for routing decisions while simultaneously reducing computational overhead in the encoder. We conducted extensive experiments on 16 VRP variants with instances ranging from 50 to 5,000 nodes. The experimental results show that Scale-Net demonstrates significant performance gains over state-of-the-art baselines across in-distribution, zero-shot, and real-world settings.

Victor Lagerkvist, Johanna Groven, Leif Eriksson

The region connection calculus (RCC) and Allen's interval algebra (IA) are two well-known NP-hard spatial-temporal qualitative reasoning problems. They are solvable in 2^(O(n log n)) time, where n is the number of variables, and IA is additionally known to be solvable in o(n)^n time. However, no improvement over exhaustive search is known for RCC, and if they are also solvable in single exponential time 2^O(n) is unknown. We investigate multiple avenues towards reaching such bounds. First, we show that branching is insufficient since there are too many non-redundant constraints. Concretely, we classify the maximum number of non-redundant constraints in RCC and IA. Algorithmically, we make two significant contributions based on dynamic programming (DP). The first algorithm runs in 4^n time and is applicable to a non-trivial, NP-hard fragment of IA, which includes the well-known interval graph sandwich problem of (Golumbic and Shamir 1993). For the richer RCC problem with 8 basic relations we use a more sophisticated approach which asymptotically matches the o(n)^n bound for IA.

Stepan Kochemazov, Oleg Zaikin, Grigorii Trofimiuk, Kirill Antonov, Alexander Semenov

In recent years, constraint solvers show increasing use in solving various open combinatorial problems, e.g., from Ramsey theory or synthesis of combinatorial designs. The similar approach can be applied to some problems related to binary linear codes, which form one of the largest families of error correcting codes used both in coding theory and in various practical applications. Thanks to a simple algebraic structure of such codes it is possible to study them using a wide range of methods. Note that even codes with the same basic parameters (length n, dimension k, minimum code distance d) can show different error correction performance, i.e., the ability to correct errors which appear in a noisy channel. In the paper, we formulate the problem of finding binary linear codes with good error correction performance as a constraint optimization problem and explore the effectiveness of modern constraint solvers on it, including SAT, MaxSAT, and CP solvers. Using the respective solvers and parallel computing, for several values of n, k, d we found the codes which are significantly better than the known in terms of their practical performance.

Markus Kirchweger, Tomáš Peitl, David Seka, Stefan Szeider

List coloring extends graph coloring by assigning each vertex a list of allowed colors. A graph is k-choosable if it can be properly colored for any choice of lists with k colors each. Deciding k-choosability is π²ₚ-complete, bipartite graphs have unbounded list chromatic number, and planar graphs (famously 4-colorable) are all 5-choosable but not all 4-choosable. To search for graphs of given choosability, we extend SAT Modulo Symmetries (SMS) with custom propagators for list coloring pruning techniques and propose a quantified Boolean (QBF) encoding for choosability. We employ a hybrid approach: pen-and-paper reasoning to optimize our formulas followed by automated case distinction by QBF solvers and SMS. Our methods yield two significant results: (1) a 27-vertex planar graph that is 4-choosable yet cannot be proven so using the combinatorial Nullstellensatz widely applied in previous work (we show this is a smallest graph with that property), and (2) the smallest graph exhibiting a gap between chromatic and list chromatic numbers for chromatic number 3.

Brendan Juba, Kuldeep S. Meel

Given a Boolean relational specification between inputs and outputs, the problem of functional synthesis is to construct a function that maps each assignment of the input to an assignment of the output such that each tuple of input and output assignments meets the specification. The past decade has witnessed significant improvement in the scalability of functional synthesis tools, allowing them to handle problems with tens of thousands of variables. A common ingredient in these approaches is their reliance on SAT solvers, thereby exploiting the breakthrough advances in SAT solving over the past three decades. While the recent techniques have been shown to perform well in practice, there is little theoretical understanding of the limitations and power of these approaches. The primary contribution of this work is to initiate a systematic theoretical investigation into the power of functional synthesis approaches that rely on NP oracles. We first show that even when small Skolem functions exist, naive bit-by-bit learning approaches fail due to the relational nature of specifications. We establish fundamental limitations of interpolation-based approaches proving that even when small Skolem functions exist, resolution-based interpolation must produce exponential-size circuits. We prove that access to an NP oracle is inherently necessary for efficient synthesis. Our main technical result shows that it is possible to use NP oracles to synthesize small Skolem functions in time polynomial in the size of the specification and the size of the smallest sufficient set of witnesses, establishing positive results for a broad class of relational specifications.

Hannes Ihalainen, Dieter Vandesande, André Schidler, Jeremias Berg, Bart Bogaerts, Matti Järvisalo

The implicit hitting set (IHS) approach offers a general framework for solving computationally hard combinatorial optimization problems declaratively. IHS iterates between a decision oracle used for extracting sources of inconsistency and an optimizer for computing so-called hitting sets (HSs) over the accumulated sources of inconsistency. While the decision oracle is language-specific, the optimizers is usually instantiated through integer programming. We explore alternative algorithmic techniques for hitting set optimization based on different ways of employing pseudo-Boolean (PB) reasoning as well as stochastic local search. We extensively evaluate the practical feasibility of the alternatives in particular in the context of pseudo-Boolean (0-1 IP) optimization as one of the most recent instantiations of IHS. Highlighting a trade-off between efficiency and reliability, while a commercial IP solver turns out to remain the most effective way to instantiate HS computations, it can cause correctness issues due to numerical instability; in fact, we show that exact HS computations instantiated via PB reasoning can be made competitive with a numerically exact IP solver. Furthermore, the use of PB reasoning as a basis for HS computations allows for obtaining certificates for the correctness of IHS computations, generally applicable to any IHS instantiation in which reasoning in the declarative language at hand can be captured in the PB-based proof format we employ.

Ramiz Gindullin, María Andreína Francisco Rodríguez

Standardized microplates are used to conduct large-scale biomedical research. The design of microplate layouts plays an essential role in handling so-called plate effects, i.e., systematic variations across the geometry of a microplate. An effective layout allows us to detect and negate plate effects. The randomized placement of controls and compounds produces layouts of limited effectiveness, so specific approaches are needed. A previously developed system, PLAID, proposed a constraint satisfaction model to construct effective plate layouts. However, PLAID does not scale well with microplate dimensions. To improve on PLAID, we propose Constraint Optimization of MicroPlate Designs (COMPD), which allows for greater flexibility and higher quality of the layouts.

Long-Hin Fung, Che Cheng, Jie-Hong Roland Jiang, Friedrich Slivovsky, Tony Tan

Dependency Quantified Boolean Formulas (DQBF) generalize QBF by explicitly specifying which universal variables each existential variable depends on, instead of relying on a linear quantifier order. The satisfiability problem of DQBF is NEXP-complete, and many hard problems can be succinctly encoded as DQBF. Recent work has revealed a strong analogy between DQBF and SAT: k-DQBF (with k existential variables) is a succinct form of k-SAT, and satisfiability is NEXP-complete for 3-DQBF but PSPACE-complete for 2-DQBF, mirroring the complexity gap between 3-SAT (NP-complete) and 2-SAT (NL-complete). Motivated by this analogy, we study the model counting problem for DQBF, denoted #DQBF. Our main theoretical result is that #2-DQBF is #EXP-complete, where #EXP is the exponential-time analogue of #P. This parallels Valiant's classical theorem stating that #2-SAT is #P-complete. As a direct application, we show that first-order model counting (FOMC) remains #EXP-complete even when restricted to a PSPACE-decidable fragment of first-order logic and domain size two. Building on recent successes in reducing 2-DQBF satisfiability to symbolic model checking, we develop a dedicated 2-DQBF model counter. Using a diverse set of crafted instances, we experimentally evaluated it against a baseline that expands 2-DQBF formulas into propositional formulas and applies propositional model counting. While the baseline worked well when each existential variable depends on few variables, our implementation scaled significantly better to larger dependency sets.