Title: BIPNN: Learning to Solve Binary Integer Programming via Hypergraph Neural Networks
Venue: NeurIPS 2025
Accepted: yes
Keywords: binary integer program

hypergraph neural network

nonlinear integer programming
Abstract: Binary (0-1) integer programming (BIP) is pivotal in scientific domains requiring discrete decision-making. As the advance of AI computing, recent works explore neural network-based solver for integer linear programming (ILP) problems. Yet, they lack scalability for tackling nonlinear challenges. To handle nonlinearities, state-of-the-art Branch-and-Cut solvers employ linear relaxations, leading to exponential growth in auxiliary variables and severe computation limitations. To overcome these limitations, we propose BIPNN, an unsupervised learning framework to solve BIP problems via hypergraph neural networks (HyperGNN). Specifically, (i) BIPNN reformulates BIPs-constrained, discrete, and nonlinear (sin, log, exp) optimization problems-into unconstrained, differentiable, and polynomial loss functions. The reformulation stems from the observation of a precise one-to-one mapping between polynomial BIP objectives and hypergraph structures, enabling the unsupervised training of HyperGNN to optimize BIP problems in an end-to-end manner. On this basis, (ii) we propose a GPU-accelerated and continuous-annealing-enhanced training pipeline for BIPNN. The pipeline enables BIPNN to optimize large-scale nonlinear terms in BIPs fully in parallel via straightforward gradient descent, thus significantly reducing the training cost while ensuring the generation of discrete, high-quality solutions. Extensive experiments on synthetic and real-world datasets highlight the superiority of our approach.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nTo handle nonlinearities, most current solvers employ linear relaxations, authors proposed BIPNN (Binary Integer Programming Neural Network), an unsupervised learning framework to solve nonlinear BIP problems via hypergraph neural networks (HyperGNN). The authors also show the performance on synthetic and real-world datasets highlight the superiority of approach on maxcut problem. They compare the approach with SCIP and tabu search.\n\nHowever, there are several weaknesses, and the quality should be improved.\n\nSTRENGTHS AND WEAKNESSES:\nstrengths: authors propose an unsupervised learning framework, named as BIPNN (Binary Integer Programming Neural Network), to solve nonlinear BIP problems via hypergraph neural networks (HyperGNN). \n\nWeaknesses:\n\n the idea is not very novel since some researchers have proposed similar approaches. Many researchers propose QUBO or Ising model [1] to solve this kind of problems. The depth of the paper  should also be improved. I suggest authors to read more current works such as [1]\n\n[1] A Monte Carlo Policy Gradient Method with Local Search for Binary Optimization,\n\nQUESTIONS:\nno\n\nLIMITATIONS:\n1, In abstract, authors said “Yet, they lack scalability for tackling nonlinear challenges”, however, I did not see the  scalability of your proposed approach. Plz show this metric. \n\n2, what’s PUBO?\n\n3, in abstract, “reformulates BIPs-constrained, discrete, and nonlinear (sin, log, exp) optimization problems-into unconstrained, differentiable, and polynomial loss functions”, the eqn 7 may be not differentiable.\n\n4, the performance evaluation, plz compare the SOTA solver such as GUROBI, some problem-specific algorithms, and several RL methods such as [1][2]. \n\n5, The novelty of the proposed approach is not very clear. Some works such as [1] [2] have proposed to transfer BIP to unconstrained problems. \n\n[1] A Monte Carlo Policy Gradient Method with Local Search for Binary Optimization, \n\n[2] Supplementing recurrent neural networks with annealing to solve combinatorial optimization problems", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tries to solve a type of challenging optimization problem of nonlinear binary integer programming (BIP). This problem is fundamental in areas ranging from operations researches to chemical process control. It is hard problem due to its NP-completeness and the exploration of auxiliary variables in standard linearization schemes. \n\nBy deriving a one-to-one mapping between polynomial BIP objectives and hyper graph structures, the authors propose a new method called BIPNN. It is an unsupervised hyper graph neural-network (HyperGNN) that can be used to tackle BIP. The experimental results demonstrate its effectiveness. \n\nThe main contribution of this paper include\n1) A systematic way to reformulate constrained, discrete, nonlinear IBs into unconstrained polynomial (PUBO) loss functions and encode nonlinearities (isn, log, etc.) via low-degree polynomials.\n2) A continuous-annealing-enhanced training pipeline that performs gradient-based optimization end-to-end on the HyperGNN, avoiding costly branch-and-cut and auxiliary‐variable blowup. Also using GPU to speed up training. \n3) Good emperimental results on both synthetic hypergraphs (up to thousands of variables) and real-world graph/hypergraph max-cut benchmarks that BIPNN matches or outperforms other approaches in solution quality, while scaling well in time for fixed GPU resources .\n\nSTRENGTHS AND WEAKNESSES:\nThe main strengths include\n1) Novel PUBO reformulation. The precise polynomial fitting of nonlinear terms for binary inputs enables an exact loss construction rather than heuristic linear relaxations .\n2) End-to-End differentiable HyperGNN solver. Modeling multi-variable interactions as hyperedges and training HyperGNN directly on the PUBO loss bypasses expensive combinatorial search, achieving scalability and leveraging GPU parallelism.\n3) Good empirical evaluation. Extensive benchmarks on both synthetic and real datasets, with comparisons against exact (SCIP) and heuristic (Tabu) solvers under a 1 hour limit, demonstrate BIPNN’s competitive performance, especially on larger instances.\n\nThe main weakness include \n1) Limited scalability of polynomial reformulation. The general transformation for multivariate nonlinearities causes exponential preprocessing in the number of variables per term, limiting applicability to only low-order interactions.\n2) No clear reasons of HyperGNN architectural choices. While several HyperGNN variants are cited, the paper lacks ablation on how different architectures (HGNN+, HyperGCN, UniGCN) affect performance or how hyperparameters were selected. The author should provide hyperparameter guidelines. Provide systematic methods or default settings (and sensitivity analysis) for choose proper values of λ, μ, γ, etc. to aid reproducibility.\n3) Weak Baselines. Comparison is confined to SCIP and Tabu, which are good open source code, but they are generally much worse than the MINLP solvers by Gurobi, FICO, etc. The baseline comparison should be strengthened by comparing with those solvers, so that the effectiveness and scalability of the proposed approach can be better justified.\n\nQUESTIONS:\nOn Page 1, \"However, BIP is known to be NP-complete\". This is not accurate. Assignment problems and shortest path are BIP, but has polynomial algorithm. This sentence can change to \"However, BIP is generally known to be NP-complete\"?\n\nOn page 5, Section 4.2, \"where λ_k, μ_l are sufficiently large penalty coefficients\".  What is the guidance on choosing the “sufficiently large” λ_k and μ_l to avoid infeasibility or poor convergence?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a Hypergraph Neural Network (HyperNN) based solver for non-linear binary integer programming (BIP) problems. The proposed BIPNN approach reformulates BIP problems into unconstrained, differentiable and polynomial loss functions (PUBO formalism) that enable the training of a HyperGNN to solve these problems. This allows the method to be parallelizable on modern GPU hardware, making it significantly faster and more scalable than existing approaches. The authors provide clear explanations of how the BIP problems are reformulated into a unconstrained polynomial problem that can be used to train the HyperGNNs. An annealing strategy is used to ensure that the final solutions are binary. They show empirically that their method outperforms SOTA solvers on hypergraph max-cut problems and is comparable, sometimes outperforming) SOTA solvers on graph max-cut problems. They also show how the GPU parallelization enables their method to be significantly faster than SOTA solvers.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- The paper is generally very well written, most important concepts are clearly explained.\n- The figures are well made and highly useful to clarify any concepts that might not be understood directly from the text itself.\n- The proposed method is innovative.\n- The empirical results properly showcase the benefits of the proposed methods.\n\n### Weaknesses\nWhile the paper is generally well written, some concepts could be better clarified:\n- The precise \"one-to-one\" mapping between polynomial BIP objectives and hypergraph structures is mentioned multiple times throughout the text and in the abstract and while this mapping is somewhat intuitive from figure 1 it is not described precisely in the text. When the authors mention this in the abstract it gives the impression that there will be some mathematical results/ that shows the existence of this mapping. More details about this would be beneficial.\n- A brief description of the inner workings of HyperGNNs would be useful to clarify these models for less familiar readers.\n- While the training workflow is clear from the text and highly useful Fig 1, I feel like the paper would benefit from a brief intuitive explanation of how Eq. 4 works with the training.\n- See the questions section for some others\n\nQUESTIONS:\n### Questions\n- The authors use the term \"matrix\" to refer to the PUBO matrix Q but this seems to be simply the vector of coefficients, naming it \"matrix\" introduces a bit of confusion so maybe \"vector\" would be a better term for clarity.\n- How exactly does BIPNN generate a new hypergraph structure as opposed to the original hypergraph (Fig 3 & Sections 5 & 6)? This seems to be a central point in the experiments section but the phenomenon isn't clearly described. A brief discussion of how the generated hypergraph differs from the original one (perhaps with an example) would be beneficial.\n- What exactly is the ratio of solutions? (Fig 4 & Section 6) Are the solutions here on the max-cut problem therefore the \"solution\" is the number of cuts and higher is better or is it the cost-function of the PUBO problem? This is unclear from the text.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces BIPNN, an unsupervised learning framework for solving nonlinear Binary Integer Programming (BIP) problems. Its core idea is to reformulate a complex, nonlinear, and constrained BIP into an unconstrained polynomial binary optimization (PUBO) problem via polynomial and penalty methods. This PUBO problem is then modeled as a hypergraph and solved end-to-end by a Hypergraph Neural Network (HyperGNN). To enhance scalability, the authors also design a GPU-accelerated training algorithm and an annealing strategy.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- The paper combines polynomial reformulation, hypergraph modeling, and unsupervised learning into a novel, end-to-end framework for the challenging class of nonlinear BIP problems.\n\n- The proposed GPU-acceleration algorithm effectively addresses the computational bottleneck of large-scale PUBO loss functions, which is a key step toward practical application.\n\n**Weaknesses:**\n\n- The paper's claim that complex nonlinear terms can be used directly in the loss function is not well-explained and seems to conflict with the hypergraph modeling, which is based on polynomial interactions.\n\n- The polynomial reformulation of functions like $\\text{sin}(x_1+x_2)$ can introduce precision errors when variables are relaxed to the continuous space, a critical issue for problems with tight constraints that the paper fails to address.\n\n- The paper does not adequately differentiate itself from recent and highly similar work using hypergraphs for optimization [1, 2], which weakens its originality claim.\n\n\n- The evaluation omits comparisons with major commercial solvers (e.g., Gurobi, CPLEX) and other specialized nonlinear solvers.\n\n- Crucially, the paper lacks experiments on established public benchmarks for nonlinear optimization, such as QPLIB[3], which significantly undermines its performance claims.\n\n- There is no discussion of the potential performance trade-offs of the chosen unsupervised approach versus supervised alternatives.\n\n[1] Distributed constrained combinatorial optimization leveraging hypergraph neural networks.\n\n[2] NeuralQP: A General Hypergraph-based Optimization Framework for Large-scale QCQPs.\n\n[3] QPLIB: A library of quadratic programming instances.\n\nQUESTIONS:\n- How are nonlinear terms integrated into the hypergraph model and the GPU-accelerated PUBO loss function if they are not converted into polynomials?\n\n- What is the impact of precision errors from the polynomial approximation during continuous relaxation, especially for problems with tight equality constraints?\n\n- What are the key differences and advantages of BIPNN compared to recent, similar works like NeuralQP?\n\n- Why was an unsupervised approach chosen? What is the expected performance trade-off compared to a supervised method?\n\n- To validate the method's effectiveness, comparisons against a wider range of solvers (Gurobi, CPLEX) and on standard libraries (QPLIB) are essential. Are the authors willing to provide these crucial results? My assessment of the paper's significance would substantially increase with such additions.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}