design branch-and-bound

This paper investigates whether the branch-and-bound (B&B) algorithm exhibits polynomial-time approximation scheme (PTAS) behavior for NP-hard combinatorial optimization problems, including the knapsack and scheduling problems. Through structural problem analysis, novel truncation strategy design, and rigorous convergence analysis, we establish—for the first time—theoretical guarantees that the standard B&B framework asymptotically generates (1−ε)-approximate solutions within polynomial time. This result fundamentally challenges the conventional view that B&B only ensures eventual optimality, and instead bridges B&B with approximation algorithms by formally extending its theoretical applicability to polynomial-time approximation. Extensive experiments on benchmark instances confirm that the proposed approach achieves arbitrary approximation accuracy ε > 0 in polynomial time, matching or surpassing the performance of specialized PTASs and state-of-the-art heuristic methods.

This work addresses integer and mixed-integer nonlinear programming (INLP/MINLP) problems, aiming to accelerate exact algorithms—particularly branch-and-bound (BB)—while rigorously preserving global optimality. We propose a unified learnable BB framework that jointly integrates supervised learning, imitation learning, and reinforcement learning into four core components: branching variable selection, cutting-plane generation, node prioritization, and parameter tuning. The framework is agnostic to variable types—supporting discrete, continuous, and hybrid structures—and is validated on real-world applications including unit commitment, vehicle routing, and hydroelectric scheduling. Key contributions include: (i) the first taxonomy of learning-augmented optimization methods organized along both solver architecture and learning paradigm dimensions; (ii) substantial convergence acceleration without compromising solution quality or optimality guarantees; and (iii) advancement toward scalable, generalizable intelligent optimization solvers.

Variable selection heuristics in branch-and-bound (B&B) for mixed-integer linear programming (MILP) suffer from low efficiency and poor generalization. Method: We propose the first model-based reinforcement learning (MBRL) framework for B&B, which learns a dynamic environment model of the B&B search process and integrates Monte Carlo tree search (MCTS) for forward-looking, adaptive branching decisions—yielding both interpretability and sample efficiency. Contribution/Results: Our approach overcomes the dual limitations of static heuristics and model-free RL in modeling capacity and data efficiency. Evaluated on four standard MILP benchmarks, it consistently outperforms state-of-the-art RL-driven branching policies, achieving significant reductions in solving time, number of explored nodes, and optimality gap. These results validate the effectiveness and scalability of model-guided planning for combinatorial optimization.

This paper addresses the monotone submodular maximization problem subject to a knapsack constraint (Submodular Knapsack Problem), aiming to deliver verifiably optimal solutions for practical applications. We propose the first dedicated branch-and-bound framework for this problem, integrating three key innovations: (i) tight upper-bound estimation leveraging submodularity, (ii) pruning rules derived directly from submodular properties, and (iii) a dynamic variable ordering strategy. Evaluated on three benchmark instance classes, our method significantly outperforms both general-purpose integer programming solvers and state-of-the-art heuristics in efficiency and scalability: it achieves an average 3.2× speedup on medium-scale instances and, for the first time, solves several previously intractable instances to optimality in polynomial time. Theoretical analysis ensures solution correctness and bound tightness, while empirical results demonstrate robust performance across diverse problem scales—establishing a new standard for exact algorithms in submodular optimization with practical relevance.

Embedding decision trees—including ensembles—into optimization problems suffers from low modeling accuracy and poor computational efficiency due to weak linear relaxations in existing mixed-integer programming (MIP) formulations. Method: We propose an ideal MIP formulation based on the union of projection polytopes, explicitly capturing tree logic via binary feature representations and extending to one-dimensional continuous features. Contribution/Results: We prove, for the first time under binary feature encoding, that allowing repeated splits on the same feature eliminates fractional extreme points in the linear relaxation. We further derive the ideal MIP characterization for univariate continuous features. Our formulation substantially tightens the linear relaxation and reduces the number of extreme points in the feasible region. On low-dimensional feature instances, average solution time decreases by an order of magnitude. This advancement significantly improves both the embeddability of tree models into optimization frameworks and their computational scalability.

This study addresses the Capacitated Profitable Tour Problem (CPTP) and its open s-t path variant by implementing and reproducing the branch-and-cut algorithm originally proposed by Jepsen et al. (2014) using the open-source solver HiGHS, thereby establishing the first fully open-source and reproducible solution framework for these problems. The implementation integrates capacity cuts, connectivity constraints, bound-based preprocessing, domain propagation, and reduced-cost variable fixing. Experimental results demonstrate that capacity cuts substantially enhance performance, increasing the number of solved instances from 52 to 64 out of 76 and reducing the search tree size by over an order of magnitude, while other components provide limited additional benefit. This work represents the first efficient branch-and-cut solver for CPTP variants built entirely on an open-source stack.

To address the low computational efficiency of solving mixed-integer bilevel linear programs (MIBLPs), this paper proposes a novel unified modeling approach based on *improving directions*: a single subproblem simultaneously verifies bilevel feasibility and generates strong valid inequalities. Theoretically, we characterize the role of improving directions in encoding the follower’s optimality conditions, establish an optimality-based relaxation hierarchy, and extend the theory of continuous cutting-plane closures to the mixed-integer bilevel setting. Algorithmically, we integrate improving-direction analysis into a branch-and-cut framework, implementing it atop the open-source solver MibS. Computational experiments demonstrate that our method substantially enhances inequality strength and overall solution performance across standard benchmark instances.

This work proposes DeepBound, a deep learning–based node selection algorithm for mixed-integer linear programming (MILP) that overcomes the limited generalization and instability of traditional handcrafted heuristics. By leveraging a multi-level feature fusion network and a pairwise training paradigm, DeepBound automatically learns optimal branching strategies from data, effectively mitigating node imbalance in branch-and-bound trees. Experimental results on three NP-hard MILP benchmarks demonstrate that DeepBound significantly outperforms both classical heuristics and existing learning-based methods, achieving substantially faster solution times, quicker discovery of high-quality feasible solutions, and strong generalization performance on large-scale, complex instances.

This paper addresses the low efficiency and poor generalizability of variable selection during early-stage branch-and-bound in online mixed-integer programming (MIP) solving. We propose an online learning method that integrates graph representation learning with Influence Branching. The constraint matrix serves as a graph-structured input, and a graph neural network models variable-constraint interactions; Thompson sampling dynamically optimizes the branching variable selection policy. Our key contributions are: (i) the first integration of Influence Branching into an online learning framework, enabling strong generalization to changes in objective functions, constraint coefficients, and problem structure; and (ii) end-to-end graph representation learning for adaptive identification of optimal subgraphs encoding discriminative features. Experiments show that our method matches state-of-the-art online approaches in solving speed while significantly improving robustness under distribution shift and scalability to larger problem instances.