稀疏整数规划的非线性边编码与列和行生成分解

中文摘要 AI 辅助

广泛的稀疏整数规划具有块结构，其中子问题通过少量共享变量相互作用。对耦合等式进行对偶化可得到可分解的拉格朗日松弛，但通常会产生对偶间隙。近期研究表明，通过对共享变量上指数大的冗余非线性一致性约束族进行对偶化，可以在保持可分解性的同时消除该间隙。我们开发了一个计算框架来利用这一思想，而无需显式构造由此产生的指数大松弛。该框架将共享变量一致性的非线性边编码与列和行生成（CRG）算法相结合，该算法通过定价生成局部整数解，并通过分离来编码约束。在完全编码和精确分离的情况下，该框架恢复精确松弛，同时保持对块的独立优化。我们引入了若干编码族，并建立了它们之间的指数级分离：广义族可能比顶点族、单项式族或反射族所需约束少指数级，但在某些实例上，单一问题特定编码即可满足时，广义族自身可能需要指数级约束。在分解的稳定集和支配集实例上的计算实验表明，在各种树拓扑和耦合强度下，CRG显著优于整体式公式。结果还表明，更丰富的编码族在计算上不一定表现更好，这凸显了编码选择作为有效分解核心问题的重要性。

英文摘要

A wide range of sparse integer programs admit a block structure in which subproblems interact through a small set of shared variables. Dualizing the linking equalities yields a decomposable Lagrangian relaxation, but generally introduces a duality gap. Recent work shows that this gap can be closed while preserving decomposability by dualizing exponentially large families of redundant nonlinear consistency constraints on the shared variables. We develop a computational framework for exploiting this idea without explicitly constructing the resulting exponentially large relaxation. Our framework combines nonlinear edge encodings of shared-variable consistency with a column-and-row generation (CRG) algorithm that generates local integer solutions by pricing and encoding constraints by separation. With complete encodings and exact separation, the framework recovers the exact relaxation while retaining independent optimization over the blocks. We introduce several encoding families and establish exponential separations among them: the Generalized family can require exponentially fewer constraints than the Vertex, Monomial, or Reflected families, yet can itself require exponentially many constraints on instances for which a single problem-specific encoding suffices. Computational experiments on decomposed stable-set and dominating-set instances show that CRG substantially outperforms a monolithic formulation across a range of tree topologies and coupling strengths. The results also show that richer encoding families need not perform better computationally, highlighting the choice of encoding as a central issue in effective decomposition.

发表机构

• Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）
• H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院 H. Milton Stewart 工业与系统工程学校）

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