AI 中文总结
本研究针对对称数值三维匹配问题，通过三类归约证明其为强NP完全、APX完全，且不存在PTAS，明确了其难解性与不可近似性，补充了相关问题的近似保持归约。
AI 中文摘要
对称数值三维匹配（SN3DM）旨在询问：三个不相交的带标签类，若具有相同的权重多重集，是否可被划分为类横截三元组，且所有三元组的目标和均相同。该问题的核心是边际对称性下的角色恢复：相同的数值目录要求仅通过关联结构重构不对称的源角色。本教程针对该对称性限制得出三个互补的难解性结果：第一部分从三维匹配问题（N3DM）进行一元多项式归约，源角色成为同一出现集合中的端口，唯一强制的填充系统为每个端口预留一个主要关联，二分图边着色恢复输出类标签，无进位混合基数编码将四个坐标打包为正整数，因此SN3DM是强NP完全问题。第二部分研究最大对称数值三维匹配问题（Max-SN3DM），仅强NP难解性本身并不排除存在多项式时间近似方案（PTAS），两个数值编译器将Petrank针对有界三维匹配（3DM）的完美完备性间隙提升至一元最大三维匹配问题（Max-N3DM），缺陷稳定性引理表明，大小为13n - d的对称匹配可产生至少大小为n - 21d的源匹配，其中n为多重集基数，d为对称缺陷，因此存在某ε > 0，将完美实例与最优值至多为完美实例(1-ε)倍的实例区分开是NP难解的，故除非P = NP，否则不存在PTAS，此外每个最大合法三元组匹配都是3近似，该问题属于APX类。第三部分提供第二部分未证明的近似保持归约，精确对编译器与单活端口分离图将三度最大三维匹配问题（Maximum 3DM）归约为一元最大对称数值三维匹配问题（Max-SN3DM），满足Max-SN3DM的最优值OPT = Gamma + Max-3DM的最优值，其中Gamma为固定偏移，最优误差一一对应传递，该L归约的常数α = 764、β = 1，因此Max-SN3DM是APX完全问题，二者不可比，已验证的是/否实例对每个构造进行了核验。
英文摘要
Symmetric Numerical Three-Dimensional Matching (SN3DM) asks whether three disjoint labeled classes with identical weight multisets can be partitioned into class-transversal triples of one common target sum. Its theme is role recovery under marginal symmetry: identical numerical catalogues force the asymmetric source roles to be reconstructed from incidence structure alone. This tutorial develops three complementary hardness results for that symmetry restriction. Part I gives a unary-polynomial reduction from N3DM. Source roles become ports in one common occurrence set, a uniquely forced filler system reserves one main incidence per port, bipartite edge coloring restores the output-class labels, and a no-carry mixed-radix encoding packs four coordinates into positive integers. Hence SN3DM is strongly NP-complete. Part II studies Max-SN3DM, for which strong NP-hardness alone does not exclude a PTAS. Two numerical compilers lift Petrank's perfect-completeness gap for bounded 3DM to unary Max-N3DM, and a defect-stability lemma shows that a symmetric matching of size 13n - d yields a source matching of size at least n - 21d, where n is the multiset cardinality, and d is a symmetric defect. Hence, for some epsilon > 0, it is NP-hard to separate perfect instances from those of optimum at most (1- epsilon) times perfect, so no PTAS exists unless P = NP. Every maximal legal triple matching is a 3-approximation, placing the problem in APX. Part III supplies the approximation-preserving reduction Part II does not claim. An exact pair compiler and a one-live-port separation map degree-three Maximum 3DM to unary Max-SN3DM with OPT(Max-SN3DM) = Gamma + OPT(Max-3DM) for a fixed offset Gamma and one-for-one optimum-error transfer. The L-reduction has constants alpha = 764 and beta = 1, so Max-SN3DM is APX-complete. The two are incomparable; worked yes / no instances audit each construction.
Comments149 pages, 7 figures
