摘 要：组合数学作为离散数学的重要分支，在算法设计中具有不可替代的作用，其优化问题的研究对提升算法效率和解决复杂计算任务至关重要。本文以组合数学理论为基础，探讨其在算法设计中的优化方法及其实际应用价值。研究旨在通过引入组合优化模型，改进传统算法的性能瓶颈，并提出一种基于动态规划与组合枚举相结合的新方法，有效降低时间复杂度并提高求解精度。通过对典型问题如旅行商问题、背包问题等的实验验证，结果表明该方法能够在多项式时间内获得近似最优解，显著优于现有算法的表现。本文的主要创新点在于将组合数学中的生成函数与图论结构相融合，为复杂优化问题提供了全新的建模思路，同时拓展了组合数学在机器学习与数据挖掘领域的应用范围。研究表明，组合数学的优化方法不仅能够增强算法的适应性，还为解决大规模离散优化问题提供了理论支持和技术保障。
关键词：组合优化；动态规划；生成函数；旅行商问题；算法效率
Optimization Problems in Algorithm Design through Combinatorial Mathematics
英文人名
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Abstract：Combinatorial mathematics, as an important branch of discrete mathematics, plays an irreplaceable role in algorithm design, and the study of its optimization problems is crucial for improving algorithm efficiency and addressing complex computational tasks. Based on the theory of combinatorial mathematics, this paper explores its optimization methods in algorithm design and their practical application values. The research aims to improve the performance bottlenecks of traditional algorithms by introducing combinatorial optimization models and proposes a novel method that combines dynamic programming with combinatorial enumeration, effectively reducing time complexity while enhancing solution accuracy. Experimental validations on typical problems such as the traveling salesman problem and the knapsack problem demonstrate that this method can obtain near-optimal solutions within polynomial time, significantly outperforming existing algorithms. The primary innovation of this paper lies in integrating generating functions from combinatorial mathematics with graph-theoretical structures, providing a new modeling approach for complex optimization problems and expanding the application scope of combinatorial mathematics in machine learning and data mining. The study shows that the optimization methods of combinatorial mathematics not only enhance algorithm adaptability but also offer theoretical support and technical guarantees for solving large-scale discrete optimization problems.
Keywords: Combination Optimization；Dynamic Programming；Generating Function；Traveling Salesman Problem；Algorithm Efficiency
目 录
引言 1
一、组合数学基础与算法优化 1
(一)组合数学的核心概念 1
(二)算法设计中的组合问题 2
(三)优化问题的数学建模 2
二、排列组合在算法优化中的应用 3
(一)排列组合的基本原理 3
(二)排列组合在搜索算法中的优化 3
(三)排列组合在动态规划中的作用 4
三、图论方法在组合优化中的实践 4
(一)图论与组合优化的关系 4
(二)最短路径问题的组合优化策略 5
(三)网络流问题的组合数学分析 5
四、高级组合技术与复杂算法优化 6
(一)分支定界法的优化机制 6
(二)贪心算法中的组合选择策略 6
(三)近似算法的组合优化改进 7
结论 7
参考文献 9
致谢 9
