使用约简规则改进最大团问题的上界
Improving Upper Bounds for the Maximum Clique Problem using Reduction Rules
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中文总结 AI 辅助
研究最大团问题中约简规则与上界函数的相互作用，提出用上限测试强化约简的方法，引出新的桁架等概念并证明相关性质，给出改进上界值的框架及实例，实验表明能显著改进上界函数并提升计算效率。
中文摘要 AI 辅助
我们研究了最大团问题（MCP）的约简规则与上界函数之间的相互作用。展示了MCP上界函数如何通过用上限测试替换局部大小条件来强化经典的核心和桁架约简。这引出了\((k,\omega^u)\)-核心、\((k,\omega^u)\)-桁架以及更一般的\((k,d,\omega^u)\)-桁架，其中参数\(d\)控制更强约简与额外计算成本之间的权衡。针对这些概念，我们证明了团保持性质、相应剥皮算法的正确性和运行时间界限。基于这些约简，我们引入了一个改进MCP上界值的通用框架，并给出了两个具体实例。在73个基准图上的计算实验表明，所提出的约简可以显著改进几个标准上界函数，并且组合多种约简方法在实践中是有益的。特别是，结构、桁架和核心约简与基于DSatur的界限相结合，通常比直接的SDP计算更快地达到SDP级上界值；在边密度低于\(0.7\)的测试图上，在每种情况下都是如此。使用桁架和核心约简与Lovász theta上界函数，我们还改进了三个未知精确团数的困难DIMACS实例的先前最佳认证整数上界值。特别是，将图\texttt{C500.9}的上界值从83提高到73，将图\texttt{C１000.9}的上界值从122提高到115，将图\texttt{C2000.9}的上界值从177提高到168。
英文摘要
We study the interaction between reduction rules and upper-bound functions for the Maximum Clique Problem (MCP). We show how MCP upper-bound functions can strengthen classical core and truss reductions by replacing local size conditions with upper-bound tests. This leads to the \((k,ω^u)\)-core, the \((k,ω^u)\)-truss, and the more general \((k,d,ω^u)\)-truss, where the parameter \(d\) controls the trade-off between stronger reductions and additional computational cost. For each of these notions, we prove clique-preservation properties, correctness of the corresponding peeling algorithm, and running-time bounds. Based on these reductions, we introduce a general framework for improving upper-bound values for MCP. We give two concrete instantiations of the framework: one that uses only the combined truss and core reductions, and one that combines the truss and core reductions with repeated applications of structions. Computational experiments on 73 benchmark graphs show that the proposed reductions can substantially improve several standard upper-bound functions and that combining multiple reduction methods can be beneficial in practice. In particular, the combination of structions, truss and core reductions with a DSatur-based bound often reached SDP-level upper-bound values faster than direct SDP computation; on the tested graphs with edge density below \(0.7\), it did so in every case. Using the truss and core reduction with the Lovász theta upper-bound function, we also improve the previously best certified integer upper-bound values for three difficult DIMACS instances whose exact clique numbers are not known. In particular, we improve upper-bound values for graph \texttt{C500.9} from 83 to 73, for graph \texttt{C1000.9} from 122 to 115, and for graph \texttt{C2000.9} from 177 to 168.
